:: GRFUNC_1 semantic presentation
theorem Th1: :: GRFUNC_1:1
canceled;
theorem Th2: :: GRFUNC_1:2
canceled;
theorem Th3: :: GRFUNC_1:3
canceled;
theorem Th4: :: GRFUNC_1:4
canceled;
theorem Th5: :: GRFUNC_1:5
canceled;
theorem Th6: :: GRFUNC_1:6
theorem Th7: :: GRFUNC_1:7
canceled;
theorem Th8: :: GRFUNC_1:8
theorem Th9: :: GRFUNC_1:9
Lemma23:
for x, y being set
for f, h being Function st (rng f) /\ (rng h) = {} & x in dom f & y in dom h holds
f . x <> h . y
theorem Th10: :: GRFUNC_1:10
canceled;
theorem Th11: :: GRFUNC_1:11
canceled;
theorem Th12: :: GRFUNC_1:12
theorem Th13: :: GRFUNC_1:13
theorem Th14: :: GRFUNC_1:14
canceled;
theorem Th15: :: GRFUNC_1:15
Lemma25:
for x, y, x1, y1 being set st [x,y] in {[x1,y1]} holds
( x = x1 & y = y1 )
theorem Th16: :: GRFUNC_1:16
theorem Th17: :: GRFUNC_1:17
canceled;
theorem Th18: :: GRFUNC_1:18
theorem Th19: :: GRFUNC_1:19
for
x1,
y1,
x2,
y2 being
set holds
(
{[x1,y1],[x2,y2]} is
Function iff (
x1 = x2 implies
y1 = y2 ) )
theorem Th20: :: GRFUNC_1:20
canceled;
theorem Th21: :: GRFUNC_1:21
canceled;
theorem Th22: :: GRFUNC_1:22
canceled;
theorem Th23: :: GRFUNC_1:23
canceled;
theorem Th24: :: GRFUNC_1:24
canceled;
theorem Th25: :: GRFUNC_1:25
theorem Th26: :: GRFUNC_1:26
theorem Th27: :: GRFUNC_1:27
theorem Th28: :: GRFUNC_1:28
theorem Th29: :: GRFUNC_1:29
theorem Th30: :: GRFUNC_1:30
theorem Th31: :: GRFUNC_1:31
theorem Th32: :: GRFUNC_1:32
Lemma31:
for x being set
for h, f, g being Function st h = f \/ g holds
( x in dom h iff ( x in dom f or x in dom g ) )
theorem Th33: :: GRFUNC_1:33
theorem Th34: :: GRFUNC_1:34
theorem Th35: :: GRFUNC_1:35
theorem Th36: :: GRFUNC_1:36
theorem Th37: :: GRFUNC_1:37
theorem Th38: :: GRFUNC_1:38
theorem Th39: :: GRFUNC_1:39
canceled;
theorem Th40: :: GRFUNC_1:40
canceled;
theorem Th41: :: GRFUNC_1:41
canceled;
theorem Th42: :: GRFUNC_1:42
canceled;
theorem Th43: :: GRFUNC_1:43
canceled;
theorem Th44: :: GRFUNC_1:44
canceled;
theorem Th45: :: GRFUNC_1:45
canceled;
theorem Th46: :: GRFUNC_1:46
theorem Th47: :: GRFUNC_1:47
theorem Th48: :: GRFUNC_1:48
canceled;
theorem Th49: :: GRFUNC_1:49
theorem Th50: :: GRFUNC_1:50
canceled;
theorem Th51: :: GRFUNC_1:51
canceled;
theorem Th52: :: GRFUNC_1:52
theorem Th53: :: GRFUNC_1:53
canceled;
theorem Th54: :: GRFUNC_1:54
canceled;
theorem Th55: :: GRFUNC_1:55
canceled;
theorem Th56: :: GRFUNC_1:56
canceled;
theorem Th57: :: GRFUNC_1:57
canceled;
theorem Th58: :: GRFUNC_1:58
canceled;
theorem Th59: :: GRFUNC_1:59
canceled;
theorem Th60: :: GRFUNC_1:60
canceled;
theorem Th61: :: GRFUNC_1:61
canceled;
theorem Th62: :: GRFUNC_1:62
canceled;
theorem Th63: :: GRFUNC_1:63
canceled;
theorem Th64: :: GRFUNC_1:64
theorem Th65: :: GRFUNC_1:65
canceled;
theorem Th66: :: GRFUNC_1:66
canceled;
theorem Th67: :: GRFUNC_1:67
theorem Th68: :: GRFUNC_1:68
canceled;
theorem Th69: :: GRFUNC_1:69
canceled;
theorem Th70: :: GRFUNC_1:70
canceled;
theorem Th71: :: GRFUNC_1:71
canceled;
theorem Th72: :: GRFUNC_1:72
canceled;
theorem Th73: :: GRFUNC_1:73
canceled;
theorem Th74: :: GRFUNC_1:74
canceled;
theorem Th75: :: GRFUNC_1:75
canceled;
theorem Th76: :: GRFUNC_1:76
canceled;
theorem Th77: :: GRFUNC_1:77
canceled;
theorem Th78: :: GRFUNC_1:78
canceled;
theorem Th79: :: GRFUNC_1:79
theorem Th80: :: GRFUNC_1:80
canceled;
theorem Th81: :: GRFUNC_1:81
canceled;
theorem Th82: :: GRFUNC_1:82
canceled;
theorem Th83: :: GRFUNC_1:83
canceled;
theorem Th84: :: GRFUNC_1:84
canceled;
theorem Th85: :: GRFUNC_1:85
canceled;
theorem Th86: :: GRFUNC_1:86
canceled;
theorem Th87: :: GRFUNC_1:87
theorem Th88: :: GRFUNC_1:88
theorem Th89: :: GRFUNC_1:89