:: JORDAN1J semantic presentation
begin
theorem Th1: :: JORDAN1J:1
for G being Go-board
for i1, i2, j1, j2 being Element of NAT st 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 < i2 & i2 <= len G holds
(G * (i1,j1)) `1 < (G * (i2,j2)) `1
proof
let G be Go-board; ::_thesis: for i1, i2, j1, j2 being Element of NAT st 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 < i2 & i2 <= len G holds
(G * (i1,j1)) `1 < (G * (i2,j2)) `1
let i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 < i2 & i2 <= len G implies (G * (i1,j1)) `1 < (G * (i2,j2)) `1 )
assume that
A1: 1 <= j1 and
A2: j1 <= width G and
A3: 1 <= j2 and
A4: j2 <= width G and
A5: 1 <= i1 and
A6: i1 < i2 and
A7: i2 <= len G ; ::_thesis: (G * (i1,j1)) `1 < (G * (i2,j2)) `1
A8: 1 <= i2 by A5, A6, XXREAL_0:2;
then (G * (i2,j1)) `1 = (G * (i2,1)) `1 by A1, A2, A7, GOBOARD5:2
.= (G * (i2,j2)) `1 by A3, A4, A7, A8, GOBOARD5:2 ;
hence (G * (i1,j1)) `1 < (G * (i2,j2)) `1 by A1, A2, A5, A6, A7, GOBOARD5:3; ::_thesis: verum
end;
theorem Th2: :: JORDAN1J:2
for G being Go-board
for i1, i2, j1, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 < j2 & j2 <= width G holds
(G * (i1,j1)) `2 < (G * (i2,j2)) `2
proof
let G be Go-board; ::_thesis: for i1, i2, j1, j2 being Element of NAT st 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 < j2 & j2 <= width G holds
(G * (i1,j1)) `2 < (G * (i2,j2)) `2
let i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 < j2 & j2 <= width G implies (G * (i1,j1)) `2 < (G * (i2,j2)) `2 )
assume that
A1: 1 <= i1 and
A2: i1 <= len G and
A3: 1 <= i2 and
A4: i2 <= len G and
A5: 1 <= j1 and
A6: j1 < j2 and
A7: j2 <= width G ; ::_thesis: (G * (i1,j1)) `2 < (G * (i2,j2)) `2
A8: 1 <= j2 by A5, A6, XXREAL_0:2;
then (G * (i1,j2)) `2 = (G * (1,j2)) `2 by A1, A2, A7, GOBOARD5:1
.= (G * (i2,j2)) `2 by A3, A4, A7, A8, GOBOARD5:1 ;
hence (G * (i1,j1)) `2 < (G * (i2,j2)) `2 by A1, A2, A5, A6, A7, GOBOARD5:4; ::_thesis: verum
end;
registration
let f be non empty FinSequence;
let g be FinSequence;
clusterf ^' g -> non empty ;
coherence
not f ^' g is empty
proof
f ^' g = f ^ ((2,(len g)) -cut g) by GRAPH_2:def_2;
hence not f ^' g is empty ; ::_thesis: verum
end;
end;
theorem Th3: :: JORDAN1J:3
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT holds (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))}
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT holds (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))}
let n be Element of NAT ; ::_thesis: (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))}
set US = (Cage (C,n)) -: (E-max (L~ (Cage (C,n))));
set LS = (Cage (C,n)) :- (E-max (L~ (Cage (C,n))));
set f = Cage (C,n);
set pW = E-max (L~ (Cage (C,n)));
set pN = N-min (L~ (Cage (C,n)));
A1: (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
A2: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A3: (Cage (C,n)) -: (E-max (L~ (Cage (C,n)))) <> {} by FINSEQ_5:47;
((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53;
then A4: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:42;
((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A2, FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A5: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by REVROT_1:3;
{(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A5, A4, TARSKI:def_2; ::_thesis: verum
end;
then A6: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_1:11;
card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_2:61;
then A7: card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by CARD_1:62;
N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:11;
then (N-max (L~ (Cage (C,n)))) `1 <= (E-max (L~ (Cage (C,n)))) `1 by A1, PSCOMP_1:24;
then A8: N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by SPRECT_2:51;
then card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by CARD_2:57;
then 2 c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A6, A7, XBOOLE_1:1;
then len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A9: rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by SPPOL_2:18;
len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) = (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, FINSEQ_5:42;
then ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) = E-max (L~ (Cage (C,n))) by A2, FINSEQ_5:45;
then A10: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A3, REVROT_1:3;
((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A2, FINSEQ_5:44
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A11: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A3, FINSEQ_6:42;
{(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A11, A10, TARSKI:def_2; ::_thesis: verum
end;
then A12: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) by CARD_1:11;
card (rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) by CARD_2:61;
then A13: card (rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by CARD_1:62;
((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A2, FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A14: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by REVROT_1:3;
(E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) ;
then A15: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A2, FINSEQ_5:46;
card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by A8, CARD_2:57;
then A16: 2 c= len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A12, A13, XBOOLE_1:1;
then A17: len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A18: rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by SPPOL_2:18;
thus (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} :: according to XBOOLE_0:def_10 ::_thesis: {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) or x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} )
assume A19: x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) ; ::_thesis: x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))}
then reconsider x1 = x as Point of (TOP-REAL 2) ;
assume A20: not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: contradiction
x in L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A19, XBOOLE_0:def_4;
then consider i1 being Element of NAT such that
A21: 1 <= i1 and
A22: i1 + 1 <= len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) and
A23: x1 in LSeg (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))),i1) by SPPOL_2:13;
A24: LSeg (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))),i1) = LSeg ((Cage (C,n)),i1) by A22, SPPOL_2:9;
x in L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A19, XBOOLE_0:def_4;
then consider i2 being Element of NAT such that
A25: 1 <= i2 and
A26: i2 + 1 <= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) and
A27: x1 in LSeg (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))),i2) by SPPOL_2:13;
set i3 = i2 -' 1;
A28: (i2 -' 1) + 1 = i2 by A25, XREAL_1:235;
then A29: 1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) <= ((i2 -' 1) + 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by A25, XREAL_1:7;
A30: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) = ((len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A2, FINSEQ_5:50;
then i2 < ((len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A26, NAT_1:13;
then i2 - 1 < (len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:19;
then A31: (i2 - 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by XREAL_1:20;
i2 - 1 >= 1 - 1 by A25, XREAL_1:9;
then A32: (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by A31, XREAL_0:def_2;
A33: LSeg (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))),i2) = LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A2, A28, SPPOL_2:10;
A34: len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) = (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, FINSEQ_5:42;
then i1 + 1 < ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 by A22, NAT_1:13;
then i1 + 1 < ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A29, XXREAL_0:2;
then A35: i1 + 1 <= (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by NAT_1:13;
A36: (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) -' 1) + 1 = (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, FINSEQ_4:21, XREAL_1:235;
(i2 -' 1) + 1 < ((len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A26, A28, A30, NAT_1:13;
then i2 -' 1 < (len (Cage (C,n))) - ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:7;
then A37: (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by XREAL_1:20;
then A38: ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 <= len (Cage (C,n)) by NAT_1:13;
now__::_thesis:_contradiction
percases ( ( i1 + 1 < (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) & i1 > 1 ) or i1 = 1 or i1 + 1 = (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) ) by A21, A35, XXREAL_0:1;
suppose ( i1 + 1 < (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) & i1 > 1 ) ; ::_thesis: contradiction
then LSeg ((Cage (C,n)),i1) misses LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A37, GOBOARD5:def_4;
then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {} by XBOOLE_0:def_7;
hence contradiction by A23, A27, A24, A33, XBOOLE_0:def_4; ::_thesis: verum
end;
supposeA39: i1 = 1 ; ::_thesis: contradiction
A40: (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) >= 0 + 2 by A17, A34, XREAL_1:7;
now__::_thesis:_contradiction
percases ( (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) > 2 or (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = 2 ) by A40, XXREAL_0:1;
suppose (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) > 2 ; ::_thesis: contradiction
then A41: i1 + 1 < (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by A39;
now__::_thesis:_contradiction
percases ( ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 < len (Cage (C,n)) or ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 = len (Cage (C,n)) ) by A38, XXREAL_0:1;
suppose ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 < len (Cage (C,n)) ; ::_thesis: contradiction
then LSeg ((Cage (C,n)),i1) misses LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A41, GOBOARD5:def_4;
then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {} by XBOOLE_0:def_7;
hence contradiction by A23, A27, A24, A33, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose ((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 = len (Cage (C,n)) ; ::_thesis: contradiction
then (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = (len (Cage (C,n))) - 1 ;
then (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = (len (Cage (C,n))) -' 1 by XREAL_0:def_2;
then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {((Cage (C,n)) /. 1)} by A39, GOBOARD7:34, REVROT_1:30;
then x1 in {((Cage (C,n)) /. 1)} by A23, A27, A24, A33, XBOOLE_0:def_4;
then x1 = (Cage (C,n)) /. 1 by TARSKI:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
hence contradiction by A20, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA42: (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = 2 ; ::_thesis: contradiction
A43: 1 + 2 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
x1 in (LSeg ((Cage (C,n)),1)) /\ (LSeg ((Cage (C,n)),(1 + 1))) by A23, A27, A24, A33, A39, A42, XBOOLE_0:def_4;
then x1 in {((Cage (C,n)) /. (1 + 1))} by A43, TOPREAL1:def_6;
then A44: x1 = (Cage (C,n)) /. (1 + 1) by TARSKI:def_1;
0 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) >= (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by A16, A34, A42, NAT_1:39;
then A45: i2 -' 1 = 0 by XREAL_1:6;
0 + 1 in dom ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_5:6;
then ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = x1 by A2, A42, A44, A45, FINSEQ_5:52;
then x1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53;
hence contradiction by A20, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA46: i1 + 1 = (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) ; ::_thesis: contradiction
(i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) >= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by NAT_1:11;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A32, XXREAL_0:2;
then ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;
then A47: (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) -' 1) + (1 + 1) <= len (Cage (C,n)) by A36;
0 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) <= (i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:7;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) = i1 + 1 by A22, A34, A46, XXREAL_0:1;
then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {((Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))} by A21, A36, A46, A47, TOPREAL1:def_6;
then x1 in {((Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))))} by A23, A27, A24, A33, XBOOLE_0:def_4;
then x1 = (Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by TARSKI:def_1
.= E-max (L~ (Cage (C,n))) by A2, FINSEQ_5:38 ;
hence contradiction by A20, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A48: ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A2, FINSEQ_5:44
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
not (Cage (C,n)) -: (E-max (L~ (Cage (C,n)))) is empty by A16, NAT_1:39;
then A49: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A48, FINSEQ_6:42;
A50: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:61;
thus {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} or x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) )
assume A51: x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))))
percases ( x = N-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A51, TARSKI:def_2;
suppose x = N-min (L~ (Cage (C,n))) ; ::_thesis: x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))))
hence x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by A9, A18, A49, A14, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))))
hence x in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by A9, A50, A18, A15, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
end;
theorem Th4: :: JORDAN1J:4
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))
set Nmi = N-min (L~ (Cage (C,n)));
set Nma = N-max (L~ (Cage (C,n)));
set Wmi = W-min (L~ (Cage (C,n)));
set Wma = W-max (L~ (Cage (C,n)));
set Ema = E-max (L~ (Cage (C,n)));
set Emi = E-min (L~ (Cage (C,n)));
set Sma = S-max (L~ (Cage (C,n)));
set Smi = S-min (L~ (Cage (C,n)));
set RotWmi = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
set RotEma = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:39;
A2: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;
then (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:72;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:71, XXREAL_0:2;
then A3: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:73, XXREAL_0:2;
then A4: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:74, XXREAL_0:2;
((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53;
then A5: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:42;
A6: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A7: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by REVROT_1:3;
{(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A7, A5, TARSKI:def_2; ::_thesis: verum
end;
then A8: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_1:11;
A9: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A10: (Cage (C,n)) -: (W-min (L~ (Cage (C,n)))) <> {} by FINSEQ_5:47;
len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, FINSEQ_5:42;
then ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) = W-min (L~ (Cage (C,n))) by A9, FINSEQ_5:45;
then A11: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A10, REVROT_1:3;
((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A9, FINSEQ_5:44
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A12: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A10, FINSEQ_6:42;
{(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A12, A11, TARSKI:def_2; ::_thesis: verum
end;
then A13: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_1:11;
card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_2:61;
then A14: card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by CARD_1:62;
A15: (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
(N-max (L~ (Cage (C,n)))) `1 <= (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:38;
then A16: (N-max (L~ (Cage (C,n)))) `1 <= E-bound (L~ (Cage (C,n))) by EUCLID:52;
(N-min (L~ (Cage (C,n)))) `1 < (N-max (L~ (Cage (C,n)))) `1 by SPRECT_2:51;
then A17: N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by A16, EUCLID:52;
then A18: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by CARD_2:57;
A19: (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:74;
then A20: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A9, A6, A3, FINSEQ_5:46, XXREAL_0:2;
A21: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A9, A6, A19, A3, FINSEQ_6:62, XXREAL_0:2;
W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13;
then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by A15, PSCOMP_1:24;
then A22: N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57;
then card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57;
then 2 c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A13, A14, XBOOLE_1:1;
then len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A23: rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by SPPOL_2:18;
A24: not E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
proof
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_5:53;
then A25: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by FINSEQ_6:42;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A9, FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A26: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3;
{(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A26, A25, TARSKI:def_2; ::_thesis: verum
end;
then A27: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_1:11;
A28: (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13;
then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by A28, PSCOMP_1:24;
then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57;
then A29: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57;
card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_2:61;
then card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by CARD_1:62;
then 2 c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A29, A27, XBOOLE_1:1;
then len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A30: rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by SPPOL_2:18;
assume E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ; ::_thesis: contradiction
then E-max (L~ (Cage (C,n))) in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A20, A23, A30, XBOOLE_0:def_4;
then E-max (L~ (Cage (C,n))) in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} by JORDAN1G:17;
then E-max (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) by A17, TARSKI:def_2;
hence contradiction by TOPREAL5:19; ::_thesis: verum
end;
A31: (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:70;
A32: (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:68;
then A33: (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A2, SPRECT_2:70, XXREAL_0:2;
then A34: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A1, A9, A4, FINSEQ_5:46, XXREAL_0:2;
A35: (E-max (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) <> 1
proof
assume A36: (E-max (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = 1 ; ::_thesis: contradiction
(N-min (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A1, A9, A33, A4, SPRECT_5:3, XXREAL_0:2
.= 1 by A2, FINSEQ_6:43 ;
hence contradiction by A32, A31, A20, A34, A36, FINSEQ_5:9; ::_thesis: verum
end;
then E-max (L~ (Cage (C,n))) in rng (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1) by A20, FINSEQ_6:78;
then A37: E-max (L~ (Cage (C,n))) in (rng (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1)) \ (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A24, XBOOLE_0:def_5;
card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_2:61;
then card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by CARD_1:62;
then 2 c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A18, A8, XBOOLE_1:1;
then len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A38: rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by SPPOL_2:18;
not W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))
proof
assume A39: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) ; ::_thesis: contradiction
((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) = ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by A6, FINSEQ_5:42
.= E-max (L~ (Cage (C,n))) by A6, FINSEQ_5:45 ;
then A40: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A39, RELAT_1:38, REVROT_1:3;
((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A6, FINSEQ_5:44
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A41: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A39, FINSEQ_6:42, RELAT_1:38;
{(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A41, A40, TARSKI:def_2; ::_thesis: verum
end;
then A42: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) by CARD_1:11;
card (rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) by CARD_2:61;
then card (rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by CARD_1:62;
then 2 c= len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by A18, A42, XBOOLE_1:1;
then len ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) by SPPOL_2:18;
then W-min (L~ (Cage (C,n))) in (L~ ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by A21, A38, A39, XBOOLE_0:def_4;
then W-min (L~ (Cage (C,n))) in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by Th3;
then W-min (L~ (Cage (C,n))) = E-max (L~ (Cage (C,n))) by A22, TARSKI:def_2;
hence contradiction by TOPREAL5:19; ::_thesis: verum
end;
then A43: W-min (L~ (Cage (C,n))) in (rng (Cage (C,n))) \ (rng ((Cage (C,n)) -: (E-max (L~ (Cage (C,n)))))) by A9, XBOOLE_0:def_5;
(Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) = (((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ^ (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1)) -: (E-max (L~ (Cage (C,n)))) by A9, FINSEQ_6:def_2
.= ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ^ ((((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1) -: (E-max (L~ (Cage (C,n))))) by A37, FINSEQ_6:67
.= ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ^ ((((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) -: (E-max (L~ (Cage (C,n))))) /^ 1) by A20, A35, FINSEQ_6:60
.= (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) :- (W-min (L~ (Cage (C,n))))) ^ ((((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) -: (E-max (L~ (Cage (C,n))))) /^ 1) by A43, FINSEQ_6:71, SPRECT_2:46
.= (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) :- (W-min (L~ (Cage (C,n))))) ^ (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /^ 1) by A9, A20, FINSEQ_6:75
.= (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) ^ (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /^ 1)) :- (W-min (L~ (Cage (C,n)))) by A21, FINSEQ_6:64
.= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n)))) by A6, FINSEQ_6:def_2 ;
hence Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n)))) by JORDAN1E:def_1; ::_thesis: verum
end;
theorem :: JORDAN1J:5
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
set p = W-min (L~ (Cage (C,n)));
A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
then (Upper_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A2, FINSEQ_5:44;
then (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92;
hence A3: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by FINSEQ_6:42; ::_thesis: W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n))
len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18;
hence W-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A3; ::_thesis: verum
end;
theorem :: JORDAN1J:6
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( W-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( W-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( W-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
set x = W-max (L~ (Cage (C,n)));
set p = W-min (L~ (Cage (C,n)));
set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:46;
A2: W-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:44;
A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A4: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46;
A5: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then (Lower_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A4, FINSEQ_5:44;
then (W-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, JORDAN1F:6, SPRECT_5:42;
then W-max (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))) by A1, A2, A5, A3, FINSEQ_6:62;
hence A6: W-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th4; ::_thesis: W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n))
len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18;
hence W-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A6; ::_thesis: verum
end;
theorem Th7: :: JORDAN1J:7
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
set x = N-min (L~ (Cage (C,n)));
set p = W-min (L~ (Cage (C,n)));
set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:46;
A2: N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:39;
A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A4: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46;
A5: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A6: (Lower_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A4, FINSEQ_5:44;
then A7: (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (N-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, JORDAN1F:6, SPRECT_5:43;
(W-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A6, A3, JORDAN1F:6, SPRECT_5:42;
then N-min (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))) by A1, A2, A5, A3, A7, FINSEQ_6:62, XXREAL_0:2;
hence A8: N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th4; ::_thesis: N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n))
len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18;
hence N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A8; ::_thesis: verum
end;
theorem :: JORDAN1J:8
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
set x = N-max (L~ (Cage (C,n)));
set p = W-min (L~ (Cage (C,n)));
set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:46;
A2: N-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:40;
A3: (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) by JORDAN1F:6;
A4: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A5: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46;
A6: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A7: (Lower_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A5, FINSEQ_5:44;
then A8: (W-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A4, JORDAN1F:6, SPRECT_5:42;
A9: (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (N-min (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A7, A4, JORDAN1F:6, SPRECT_5:43;
percases ( N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) <> E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) or N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) ) ;
suppose N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) <> E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) ; ::_thesis: ( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
then (W-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A7, A3, A4, A9, SPRECT_5:44, XXREAL_0:2;
then N-max (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) :- (W-min (L~ (Cage (C,n))))) by A1, A2, A6, A4, A8, FINSEQ_6:62, XXREAL_0:2;
hence A10: N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th4; ::_thesis: N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n))
len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18;
hence N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A10; ::_thesis: verum
end;
supposeA11: N-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) ; ::_thesis: ( N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
A12: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A13: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
hence A14: N-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A4, A11, A13, A12, FINSEQ_5:46; ::_thesis: N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n))
len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18;
hence N-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A14; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:9
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) )
set x = E-max (L~ (Cage (C,n)));
set p = W-min (L~ (Cage (C,n)));
A1: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
hence A3: E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A2, A1, FINSEQ_5:46; ::_thesis: E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n))
len (Upper_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by SPPOL_2:18;
hence E-max (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by A3; ::_thesis: verum
end;
theorem :: JORDAN1J:10
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( E-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( E-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( E-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
set p = E-max (L~ (Cage (C,n)));
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_2;
hence A1: E-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by FINSEQ_6:61; ::_thesis: E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n))
len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18;
hence E-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A1; ::_thesis: verum
end;
theorem :: JORDAN1J:11
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( E-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( E-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( E-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
set x = E-min (L~ (Cage (C,n)));
set p = E-max (L~ (Cage (C,n)));
set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1: rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43;
A2: E-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:45;
A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A4: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
A5: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
then (Upper_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A4, FINSEQ_5:44;
then (E-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A3, JORDAN1F:5, SPRECT_5:26;
then E-min (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) by A1, A2, A5, A3, FINSEQ_6:62;
hence A6: E-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by JORDAN1E:def_2; ::_thesis: E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n))
len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18;
hence E-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A6; ::_thesis: verum
end;
theorem Th12: :: JORDAN1J:12
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
set x = S-max (L~ (Cage (C,n)));
set p = E-max (L~ (Cage (C,n)));
set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1: rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43;
A2: S-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:42;
A3: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A4: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
A5: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
then A6: (Upper_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A4, FINSEQ_5:44;
then A7: (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (S-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A3, JORDAN1F:5, SPRECT_5:27;
(E-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A6, A3, JORDAN1F:5, SPRECT_5:26;
then S-max (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) by A1, A2, A5, A3, A7, FINSEQ_6:62, XXREAL_0:2;
hence A8: S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by JORDAN1E:def_2; ::_thesis: S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n))
len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18;
hence S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A8; ::_thesis: verum
end;
theorem :: JORDAN1J:13
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
set x = S-min (L~ (Cage (C,n)));
set p = E-max (L~ (Cage (C,n)));
set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1: rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43;
A2: S-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:41;
A3: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
A4: L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A5: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
A6: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
then A7: (Upper_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A5, FINSEQ_5:44;
then A8: (E-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A4, JORDAN1F:5, SPRECT_5:26;
A9: (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (S-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A7, A4, JORDAN1F:5, SPRECT_5:27;
percases ( S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) <> W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) or S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) = W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) ) ;
suppose S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) <> W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) ; ::_thesis: ( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
then (E-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A7, A3, A4, A9, SPRECT_5:28, XXREAL_0:2;
then S-min (L~ (Cage (C,n))) in rng ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) by A1, A2, A6, A4, A8, FINSEQ_6:62, XXREAL_0:2;
hence A10: S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by JORDAN1E:def_2; ::_thesis: S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n))
len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18;
hence S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A10; ::_thesis: verum
end;
supposeA11: S-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) = W-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) ; ::_thesis: ( S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
(Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) by JORDAN1F:8;
hence A12: S-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by A4, A11, REVROT_1:3; ::_thesis: S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n))
len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18;
hence S-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A12; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:14
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) & W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) )
set p = W-min (L~ (Cage (C,n)));
(Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) by JORDAN1F:8;
hence A1: W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by REVROT_1:3; ::_thesis: W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n))
len (Lower_Seq (C,n)) >= 2 by TOPREAL1:def_8;
then rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by SPPOL_2:18;
hence W-min (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by A1; ::_thesis: verum
end;
theorem Th15: :: JORDAN1J:15
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & N-min Y in X holds
N-min X = N-min Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & N-min Y in X implies N-min X = N-min Y )
assume that
A1: X c= Y and
A2: N-min Y in X ; ::_thesis: N-min X = N-min Y
A3: N-bound X >= (N-min Y) `2 by A2, PSCOMP_1:24;
A4: (N-min X) `2 = N-bound X by EUCLID:52;
A5: (N-min Y) `2 = N-bound Y by EUCLID:52;
A6: N-bound X <= N-bound Y by A1, PSCOMP_1:66;
then A7: N-bound X = N-bound Y by A5, A3, XXREAL_0:1;
N-min Y in N-most X by A2, A6, A5, A3, SPRECT_2:10, XXREAL_0:1;
then A8: (N-min X) `1 <= (N-min Y) `1 by PSCOMP_1:39;
N-min X in X by SPRECT_1:11;
then N-min X in N-most Y by A1, A6, A4, A5, A3, SPRECT_2:10, XXREAL_0:1;
then (N-min X) `1 >= (N-min Y) `1 by PSCOMP_1:39;
then (N-min X) `1 = (N-min Y) `1 by A8, XXREAL_0:1;
hence N-min X = N-min Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum
end;
theorem Th16: :: JORDAN1J:16
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & N-max Y in X holds
N-max X = N-max Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & N-max Y in X implies N-max X = N-max Y )
assume that
A1: X c= Y and
A2: N-max Y in X ; ::_thesis: N-max X = N-max Y
A3: N-bound X >= (N-max Y) `2 by A2, PSCOMP_1:24;
A4: (N-max X) `2 = N-bound X by EUCLID:52;
A5: (N-max Y) `2 = N-bound Y by EUCLID:52;
A6: N-bound X <= N-bound Y by A1, PSCOMP_1:66;
then A7: N-bound X = N-bound Y by A5, A3, XXREAL_0:1;
N-max Y in N-most X by A2, A6, A5, A3, SPRECT_2:10, XXREAL_0:1;
then A8: (N-max X) `1 >= (N-max Y) `1 by PSCOMP_1:39;
N-max X in X by SPRECT_1:11;
then N-max X in N-most Y by A1, A6, A4, A5, A3, SPRECT_2:10, XXREAL_0:1;
then (N-max X) `1 <= (N-max Y) `1 by PSCOMP_1:39;
then (N-max X) `1 = (N-max Y) `1 by A8, XXREAL_0:1;
hence N-max X = N-max Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum
end;
theorem Th17: :: JORDAN1J:17
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & E-min Y in X holds
E-min X = E-min Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & E-min Y in X implies E-min X = E-min Y )
assume that
A1: X c= Y and
A2: E-min Y in X ; ::_thesis: E-min X = E-min Y
A3: E-bound X >= (E-min Y) `1 by A2, PSCOMP_1:24;
A4: (E-min X) `1 = E-bound X by EUCLID:52;
A5: (E-min Y) `1 = E-bound Y by EUCLID:52;
A6: E-bound X <= E-bound Y by A1, PSCOMP_1:67;
then A7: E-bound X = E-bound Y by A5, A3, XXREAL_0:1;
E-min Y in E-most X by A2, A6, A5, A3, SPRECT_2:13, XXREAL_0:1;
then A8: (E-min X) `2 <= (E-min Y) `2 by PSCOMP_1:47;
E-min X in X by SPRECT_1:14;
then E-min X in E-most Y by A1, A6, A4, A5, A3, SPRECT_2:13, XXREAL_0:1;
then (E-min X) `2 >= (E-min Y) `2 by PSCOMP_1:47;
then (E-min X) `2 = (E-min Y) `2 by A8, XXREAL_0:1;
hence E-min X = E-min Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum
end;
theorem Th18: :: JORDAN1J:18
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & E-max Y in X holds
E-max X = E-max Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & E-max Y in X implies E-max X = E-max Y )
assume that
A1: X c= Y and
A2: E-max Y in X ; ::_thesis: E-max X = E-max Y
A3: E-bound X >= (E-max Y) `1 by A2, PSCOMP_1:24;
A4: (E-max X) `1 = E-bound X by EUCLID:52;
A5: (E-max Y) `1 = E-bound Y by EUCLID:52;
A6: E-bound X <= E-bound Y by A1, PSCOMP_1:67;
then A7: E-bound X = E-bound Y by A5, A3, XXREAL_0:1;
E-max Y in E-most X by A2, A6, A5, A3, SPRECT_2:13, XXREAL_0:1;
then A8: (E-max X) `2 >= (E-max Y) `2 by PSCOMP_1:47;
E-max X in X by SPRECT_1:14;
then E-max X in E-most Y by A1, A6, A4, A5, A3, SPRECT_2:13, XXREAL_0:1;
then (E-max X) `2 <= (E-max Y) `2 by PSCOMP_1:47;
then (E-max X) `2 = (E-max Y) `2 by A8, XXREAL_0:1;
hence E-max X = E-max Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum
end;
theorem Th19: :: JORDAN1J:19
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & S-min Y in X holds
S-min X = S-min Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & S-min Y in X implies S-min X = S-min Y )
assume that
A1: X c= Y and
A2: S-min Y in X ; ::_thesis: S-min X = S-min Y
A3: S-bound X <= (S-min Y) `2 by A2, PSCOMP_1:24;
A4: (S-min X) `2 = S-bound X by EUCLID:52;
A5: (S-min Y) `2 = S-bound Y by EUCLID:52;
A6: S-bound X >= S-bound Y by A1, PSCOMP_1:68;
then A7: S-bound X = S-bound Y by A5, A3, XXREAL_0:1;
S-min Y in S-most X by A2, A6, A5, A3, SPRECT_2:11, XXREAL_0:1;
then A8: (S-min X) `1 <= (S-min Y) `1 by PSCOMP_1:55;
S-min X in X by SPRECT_1:12;
then S-min X in S-most Y by A1, A6, A4, A5, A3, SPRECT_2:11, XXREAL_0:1;
then (S-min X) `1 >= (S-min Y) `1 by PSCOMP_1:55;
then (S-min X) `1 = (S-min Y) `1 by A8, XXREAL_0:1;
hence S-min X = S-min Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum
end;
theorem Th20: :: JORDAN1J:20
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & S-max Y in X holds
S-max X = S-max Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & S-max Y in X implies S-max X = S-max Y )
assume that
A1: X c= Y and
A2: S-max Y in X ; ::_thesis: S-max X = S-max Y
A3: S-bound X <= (S-max Y) `2 by A2, PSCOMP_1:24;
A4: (S-max X) `2 = S-bound X by EUCLID:52;
A5: (S-max Y) `2 = S-bound Y by EUCLID:52;
A6: S-bound X >= S-bound Y by A1, PSCOMP_1:68;
then A7: S-bound X = S-bound Y by A5, A3, XXREAL_0:1;
S-max Y in S-most X by A2, A6, A5, A3, SPRECT_2:11, XXREAL_0:1;
then A8: (S-max X) `1 >= (S-max Y) `1 by PSCOMP_1:55;
S-max X in X by SPRECT_1:12;
then S-max X in S-most Y by A1, A6, A4, A5, A3, SPRECT_2:11, XXREAL_0:1;
then (S-max X) `1 <= (S-max Y) `1 by PSCOMP_1:55;
then (S-max X) `1 = (S-max Y) `1 by A8, XXREAL_0:1;
hence S-max X = S-max Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum
end;
theorem Th21: :: JORDAN1J:21
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & W-min Y in X holds
W-min X = W-min Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & W-min Y in X implies W-min X = W-min Y )
assume that
A1: X c= Y and
A2: W-min Y in X ; ::_thesis: W-min X = W-min Y
A3: W-bound X <= (W-min Y) `1 by A2, PSCOMP_1:24;
A4: (W-min X) `1 = W-bound X by EUCLID:52;
A5: (W-min Y) `1 = W-bound Y by EUCLID:52;
A6: W-bound X >= W-bound Y by A1, PSCOMP_1:69;
then A7: W-bound X = W-bound Y by A5, A3, XXREAL_0:1;
W-min Y in W-most X by A2, A6, A5, A3, SPRECT_2:12, XXREAL_0:1;
then A8: (W-min X) `2 <= (W-min Y) `2 by PSCOMP_1:31;
W-min X in X by SPRECT_1:13;
then W-min X in W-most Y by A1, A6, A4, A5, A3, SPRECT_2:12, XXREAL_0:1;
then (W-min X) `2 >= (W-min Y) `2 by PSCOMP_1:31;
then (W-min X) `2 = (W-min Y) `2 by A8, XXREAL_0:1;
hence W-min X = W-min Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum
end;
theorem Th22: :: JORDAN1J:22
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & W-max Y in X holds
W-max X = W-max Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & W-max Y in X implies W-max X = W-max Y )
assume that
A1: X c= Y and
A2: W-max Y in X ; ::_thesis: W-max X = W-max Y
A3: W-bound X <= (W-max Y) `1 by A2, PSCOMP_1:24;
A4: (W-max X) `1 = W-bound X by EUCLID:52;
A5: (W-max Y) `1 = W-bound Y by EUCLID:52;
A6: W-bound X >= W-bound Y by A1, PSCOMP_1:69;
then A7: W-bound X = W-bound Y by A5, A3, XXREAL_0:1;
W-max Y in W-most X by A2, A6, A5, A3, SPRECT_2:12, XXREAL_0:1;
then A8: (W-max X) `2 >= (W-max Y) `2 by PSCOMP_1:31;
W-max X in X by SPRECT_1:13;
then W-max X in W-most Y by A1, A6, A4, A5, A3, SPRECT_2:12, XXREAL_0:1;
then (W-max X) `2 <= (W-max Y) `2 by PSCOMP_1:31;
then (W-max X) `2 = (W-max Y) `2 by A8, XXREAL_0:1;
hence W-max X = W-max Y by A4, A5, A7, TOPREAL3:6; ::_thesis: verum
end;
theorem Th23: :: JORDAN1J:23
for X, Y being non empty compact Subset of (TOP-REAL 2) st N-bound X <= N-bound Y holds
N-bound (X \/ Y) = N-bound Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( N-bound X <= N-bound Y implies N-bound (X \/ Y) = N-bound Y )
assume N-bound X <= N-bound Y ; ::_thesis: N-bound (X \/ Y) = N-bound Y
then max ((N-bound X),(N-bound Y)) = N-bound Y by XXREAL_0:def_10;
hence N-bound (X \/ Y) = N-bound Y by SPRECT_1:49; ::_thesis: verum
end;
theorem Th24: :: JORDAN1J:24
for X, Y being non empty compact Subset of (TOP-REAL 2) st E-bound X <= E-bound Y holds
E-bound (X \/ Y) = E-bound Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( E-bound X <= E-bound Y implies E-bound (X \/ Y) = E-bound Y )
assume E-bound X <= E-bound Y ; ::_thesis: E-bound (X \/ Y) = E-bound Y
then max ((E-bound X),(E-bound Y)) = E-bound Y by XXREAL_0:def_10;
hence E-bound (X \/ Y) = E-bound Y by SPRECT_1:50; ::_thesis: verum
end;
theorem Th25: :: JORDAN1J:25
for X, Y being non empty compact Subset of (TOP-REAL 2) st S-bound X <= S-bound Y holds
S-bound (X \/ Y) = S-bound X
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( S-bound X <= S-bound Y implies S-bound (X \/ Y) = S-bound X )
assume S-bound X <= S-bound Y ; ::_thesis: S-bound (X \/ Y) = S-bound X
then min ((S-bound X),(S-bound Y)) = S-bound X by XXREAL_0:def_9;
hence S-bound (X \/ Y) = S-bound X by SPRECT_1:48; ::_thesis: verum
end;
theorem Th26: :: JORDAN1J:26
for X, Y being non empty compact Subset of (TOP-REAL 2) st W-bound X <= W-bound Y holds
W-bound (X \/ Y) = W-bound X
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( W-bound X <= W-bound Y implies W-bound (X \/ Y) = W-bound X )
assume W-bound X <= W-bound Y ; ::_thesis: W-bound (X \/ Y) = W-bound X
then min ((W-bound X),(W-bound Y)) = W-bound X by XXREAL_0:def_9;
hence W-bound (X \/ Y) = W-bound X by SPRECT_1:47; ::_thesis: verum
end;
theorem :: JORDAN1J:27
for X, Y being non empty compact Subset of (TOP-REAL 2) st N-bound X < N-bound Y holds
N-min (X \/ Y) = N-min Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( N-bound X < N-bound Y implies N-min (X \/ Y) = N-min Y )
A1: (N-min (X \/ Y)) `2 = N-bound (X \/ Y) by EUCLID:52;
A2: X \/ Y is compact by COMPTS_1:10;
then A3: N-min (X \/ Y) in X \/ Y by SPRECT_1:11;
A4: N-min Y in Y by SPRECT_1:11;
A5: (N-min Y) `2 = N-bound Y by EUCLID:52;
assume A6: N-bound X < N-bound Y ; ::_thesis: N-min (X \/ Y) = N-min Y
then A7: N-bound (X \/ Y) = N-bound Y by Th23;
Y c= X \/ Y by XBOOLE_1:7;
then N-min Y in N-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:10;
then A8: (N-min (X \/ Y)) `1 <= (N-min Y) `1 by A2, PSCOMP_1:39;
percases ( N-min (X \/ Y) in Y or N-min (X \/ Y) in X ) by A3, XBOOLE_0:def_3;
suppose N-min (X \/ Y) in Y ; ::_thesis: N-min (X \/ Y) = N-min Y
then N-min (X \/ Y) in N-most Y by A6, A1, Th23, SPRECT_2:10;
then (N-min (X \/ Y)) `1 >= (N-min Y) `1 by PSCOMP_1:39;
then (N-min (X \/ Y)) `1 = (N-min Y) `1 by A8, XXREAL_0:1;
hence N-min (X \/ Y) = N-min Y by A6, A1, A5, Th23, TOPREAL3:6; ::_thesis: verum
end;
suppose N-min (X \/ Y) in X ; ::_thesis: N-min (X \/ Y) = N-min Y
hence N-min (X \/ Y) = N-min Y by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:28
for X, Y being non empty compact Subset of (TOP-REAL 2) st N-bound X < N-bound Y holds
N-max (X \/ Y) = N-max Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( N-bound X < N-bound Y implies N-max (X \/ Y) = N-max Y )
A1: (N-max (X \/ Y)) `2 = N-bound (X \/ Y) by EUCLID:52;
A2: X \/ Y is compact by COMPTS_1:10;
then A3: N-max (X \/ Y) in X \/ Y by SPRECT_1:11;
A4: N-max Y in Y by SPRECT_1:11;
A5: (N-max Y) `2 = N-bound Y by EUCLID:52;
assume A6: N-bound X < N-bound Y ; ::_thesis: N-max (X \/ Y) = N-max Y
then A7: N-bound (X \/ Y) = N-bound Y by Th23;
Y c= X \/ Y by XBOOLE_1:7;
then N-max Y in N-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:10;
then A8: (N-max (X \/ Y)) `1 >= (N-max Y) `1 by A2, PSCOMP_1:39;
percases ( N-max (X \/ Y) in Y or N-max (X \/ Y) in X ) by A3, XBOOLE_0:def_3;
suppose N-max (X \/ Y) in Y ; ::_thesis: N-max (X \/ Y) = N-max Y
then N-max (X \/ Y) in N-most Y by A6, A1, Th23, SPRECT_2:10;
then (N-max (X \/ Y)) `1 <= (N-max Y) `1 by PSCOMP_1:39;
then (N-max (X \/ Y)) `1 = (N-max Y) `1 by A8, XXREAL_0:1;
hence N-max (X \/ Y) = N-max Y by A6, A1, A5, Th23, TOPREAL3:6; ::_thesis: verum
end;
suppose N-max (X \/ Y) in X ; ::_thesis: N-max (X \/ Y) = N-max Y
hence N-max (X \/ Y) = N-max Y by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:29
for X, Y being non empty compact Subset of (TOP-REAL 2) st E-bound X < E-bound Y holds
E-min (X \/ Y) = E-min Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( E-bound X < E-bound Y implies E-min (X \/ Y) = E-min Y )
A1: (E-min (X \/ Y)) `1 = E-bound (X \/ Y) by EUCLID:52;
A2: X \/ Y is compact by COMPTS_1:10;
then A3: E-min (X \/ Y) in X \/ Y by SPRECT_1:14;
A4: E-min Y in Y by SPRECT_1:14;
A5: (E-min Y) `1 = E-bound Y by EUCLID:52;
assume A6: E-bound X < E-bound Y ; ::_thesis: E-min (X \/ Y) = E-min Y
then A7: E-bound (X \/ Y) = E-bound Y by Th24;
Y c= X \/ Y by XBOOLE_1:7;
then E-min Y in E-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:13;
then A8: (E-min (X \/ Y)) `2 <= (E-min Y) `2 by A2, PSCOMP_1:47;
percases ( E-min (X \/ Y) in Y or E-min (X \/ Y) in X ) by A3, XBOOLE_0:def_3;
suppose E-min (X \/ Y) in Y ; ::_thesis: E-min (X \/ Y) = E-min Y
then E-min (X \/ Y) in E-most Y by A6, A1, Th24, SPRECT_2:13;
then (E-min (X \/ Y)) `2 >= (E-min Y) `2 by PSCOMP_1:47;
then (E-min (X \/ Y)) `2 = (E-min Y) `2 by A8, XXREAL_0:1;
hence E-min (X \/ Y) = E-min Y by A6, A1, A5, Th24, TOPREAL3:6; ::_thesis: verum
end;
suppose E-min (X \/ Y) in X ; ::_thesis: E-min (X \/ Y) = E-min Y
hence E-min (X \/ Y) = E-min Y by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:30
for X, Y being non empty compact Subset of (TOP-REAL 2) st E-bound X < E-bound Y holds
E-max (X \/ Y) = E-max Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( E-bound X < E-bound Y implies E-max (X \/ Y) = E-max Y )
A1: (E-max (X \/ Y)) `1 = E-bound (X \/ Y) by EUCLID:52;
A2: X \/ Y is compact by COMPTS_1:10;
then A3: E-max (X \/ Y) in X \/ Y by SPRECT_1:14;
A4: E-max Y in Y by SPRECT_1:14;
A5: (E-max Y) `1 = E-bound Y by EUCLID:52;
assume A6: E-bound X < E-bound Y ; ::_thesis: E-max (X \/ Y) = E-max Y
then A7: E-bound (X \/ Y) = E-bound Y by Th24;
Y c= X \/ Y by XBOOLE_1:7;
then E-max Y in E-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:13;
then A8: (E-max (X \/ Y)) `2 >= (E-max Y) `2 by A2, PSCOMP_1:47;
percases ( E-max (X \/ Y) in Y or E-max (X \/ Y) in X ) by A3, XBOOLE_0:def_3;
suppose E-max (X \/ Y) in Y ; ::_thesis: E-max (X \/ Y) = E-max Y
then E-max (X \/ Y) in E-most Y by A6, A1, Th24, SPRECT_2:13;
then (E-max (X \/ Y)) `2 <= (E-max Y) `2 by PSCOMP_1:47;
then (E-max (X \/ Y)) `2 = (E-max Y) `2 by A8, XXREAL_0:1;
hence E-max (X \/ Y) = E-max Y by A6, A1, A5, Th24, TOPREAL3:6; ::_thesis: verum
end;
suppose E-max (X \/ Y) in X ; ::_thesis: E-max (X \/ Y) = E-max Y
hence E-max (X \/ Y) = E-max Y by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:31
for X, Y being non empty compact Subset of (TOP-REAL 2) st S-bound X < S-bound Y holds
S-min (X \/ Y) = S-min X
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( S-bound X < S-bound Y implies S-min (X \/ Y) = S-min X )
A1: (S-min (X \/ Y)) `2 = S-bound (X \/ Y) by EUCLID:52;
A2: X \/ Y is compact by COMPTS_1:10;
then A3: S-min (X \/ Y) in X \/ Y by SPRECT_1:12;
A4: S-min X in X by SPRECT_1:12;
A5: (S-min X) `2 = S-bound X by EUCLID:52;
assume A6: S-bound X < S-bound Y ; ::_thesis: S-min (X \/ Y) = S-min X
then A7: S-bound (X \/ Y) = S-bound X by Th25;
X c= X \/ Y by XBOOLE_1:7;
then S-min X in S-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:11;
then A8: (S-min (X \/ Y)) `1 <= (S-min X) `1 by A2, PSCOMP_1:55;
percases ( S-min (X \/ Y) in X or S-min (X \/ Y) in Y ) by A3, XBOOLE_0:def_3;
suppose S-min (X \/ Y) in X ; ::_thesis: S-min (X \/ Y) = S-min X
then S-min (X \/ Y) in S-most X by A6, A1, Th25, SPRECT_2:11;
then (S-min (X \/ Y)) `1 >= (S-min X) `1 by PSCOMP_1:55;
then (S-min (X \/ Y)) `1 = (S-min X) `1 by A8, XXREAL_0:1;
hence S-min (X \/ Y) = S-min X by A6, A1, A5, Th25, TOPREAL3:6; ::_thesis: verum
end;
suppose S-min (X \/ Y) in Y ; ::_thesis: S-min (X \/ Y) = S-min X
hence S-min (X \/ Y) = S-min X by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:32
for X, Y being non empty compact Subset of (TOP-REAL 2) st S-bound X < S-bound Y holds
S-max (X \/ Y) = S-max X
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( S-bound X < S-bound Y implies S-max (X \/ Y) = S-max X )
A1: (S-max (X \/ Y)) `2 = S-bound (X \/ Y) by EUCLID:52;
A2: X \/ Y is compact by COMPTS_1:10;
then A3: S-max (X \/ Y) in X \/ Y by SPRECT_1:12;
A4: S-max X in X by SPRECT_1:12;
A5: (S-max X) `2 = S-bound X by EUCLID:52;
assume A6: S-bound X < S-bound Y ; ::_thesis: S-max (X \/ Y) = S-max X
then A7: S-bound (X \/ Y) = S-bound X by Th25;
X c= X \/ Y by XBOOLE_1:7;
then S-max X in S-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:11;
then A8: (S-max (X \/ Y)) `1 >= (S-max X) `1 by A2, PSCOMP_1:55;
percases ( S-max (X \/ Y) in X or S-max (X \/ Y) in Y ) by A3, XBOOLE_0:def_3;
suppose S-max (X \/ Y) in X ; ::_thesis: S-max (X \/ Y) = S-max X
then S-max (X \/ Y) in S-most X by A6, A1, Th25, SPRECT_2:11;
then (S-max (X \/ Y)) `1 <= (S-max X) `1 by PSCOMP_1:55;
then (S-max (X \/ Y)) `1 = (S-max X) `1 by A8, XXREAL_0:1;
hence S-max (X \/ Y) = S-max X by A6, A1, A5, Th25, TOPREAL3:6; ::_thesis: verum
end;
suppose S-max (X \/ Y) in Y ; ::_thesis: S-max (X \/ Y) = S-max X
hence S-max (X \/ Y) = S-max X by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum
end;
end;
end;
theorem Th33: :: JORDAN1J:33
for X, Y being non empty compact Subset of (TOP-REAL 2) st W-bound X < W-bound Y holds
W-min (X \/ Y) = W-min X
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( W-bound X < W-bound Y implies W-min (X \/ Y) = W-min X )
A1: (W-min (X \/ Y)) `1 = W-bound (X \/ Y) by EUCLID:52;
A2: X \/ Y is compact by COMPTS_1:10;
then A3: W-min (X \/ Y) in X \/ Y by SPRECT_1:13;
A4: W-min X in X by SPRECT_1:13;
A5: (W-min X) `1 = W-bound X by EUCLID:52;
assume A6: W-bound X < W-bound Y ; ::_thesis: W-min (X \/ Y) = W-min X
then A7: W-bound (X \/ Y) = W-bound X by Th26;
X c= X \/ Y by XBOOLE_1:7;
then W-min X in W-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:12;
then A8: (W-min (X \/ Y)) `2 <= (W-min X) `2 by A2, PSCOMP_1:31;
percases ( W-min (X \/ Y) in X or W-min (X \/ Y) in Y ) by A3, XBOOLE_0:def_3;
suppose W-min (X \/ Y) in X ; ::_thesis: W-min (X \/ Y) = W-min X
then W-min (X \/ Y) in W-most X by A6, A1, Th26, SPRECT_2:12;
then (W-min (X \/ Y)) `2 >= (W-min X) `2 by PSCOMP_1:31;
then (W-min (X \/ Y)) `2 = (W-min X) `2 by A8, XXREAL_0:1;
hence W-min (X \/ Y) = W-min X by A6, A1, A5, Th26, TOPREAL3:6; ::_thesis: verum
end;
suppose W-min (X \/ Y) in Y ; ::_thesis: W-min (X \/ Y) = W-min X
hence W-min (X \/ Y) = W-min X by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:34
for X, Y being non empty compact Subset of (TOP-REAL 2) st W-bound X < W-bound Y holds
W-max (X \/ Y) = W-max X
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( W-bound X < W-bound Y implies W-max (X \/ Y) = W-max X )
A1: (W-max (X \/ Y)) `1 = W-bound (X \/ Y) by EUCLID:52;
A2: X \/ Y is compact by COMPTS_1:10;
then A3: W-max (X \/ Y) in X \/ Y by SPRECT_1:13;
A4: W-max X in X by SPRECT_1:13;
A5: (W-max X) `1 = W-bound X by EUCLID:52;
assume A6: W-bound X < W-bound Y ; ::_thesis: W-max (X \/ Y) = W-max X
then A7: W-bound (X \/ Y) = W-bound X by Th26;
X c= X \/ Y by XBOOLE_1:7;
then W-max X in W-most (X \/ Y) by A2, A7, A5, A4, SPRECT_2:12;
then A8: (W-max (X \/ Y)) `2 >= (W-max X) `2 by A2, PSCOMP_1:31;
percases ( W-max (X \/ Y) in X or W-max (X \/ Y) in Y ) by A3, XBOOLE_0:def_3;
suppose W-max (X \/ Y) in X ; ::_thesis: W-max (X \/ Y) = W-max X
then W-max (X \/ Y) in W-most X by A6, A1, Th26, SPRECT_2:12;
then (W-max (X \/ Y)) `2 <= (W-max X) `2 by PSCOMP_1:31;
then (W-max (X \/ Y)) `2 = (W-max X) `2 by A8, XXREAL_0:1;
hence W-max (X \/ Y) = W-max X by A6, A1, A5, Th26, TOPREAL3:6; ::_thesis: verum
end;
suppose W-max (X \/ Y) in Y ; ::_thesis: W-max (X \/ Y) = W-max X
hence W-max (X \/ Y) = W-max X by A6, A7, A1, PSCOMP_1:24; ::_thesis: verum
end;
end;
end;
theorem Th35: :: JORDAN1J:35
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f holds
(L_Cut (f,p)) /. (len (L_Cut (f,p))) = f /. (len f)
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f holds
(L_Cut (f,p)) /. (len (L_Cut (f,p))) = f /. (len f)
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f implies (L_Cut (f,p)) /. (len (L_Cut (f,p))) = f /. (len f) )
assume that
A1: f is being_S-Seq and
A2: p in L~ f ; ::_thesis: (L_Cut (f,p)) /. (len (L_Cut (f,p))) = f /. (len f)
A3: len f in dom f by A1, FINSEQ_5:6;
L_Cut (f,p) <> {} by A2, JORDAN1E:3;
then len (L_Cut (f,p)) in dom (L_Cut (f,p)) by FINSEQ_5:6;
hence (L_Cut (f,p)) /. (len (L_Cut (f,p))) = (L_Cut (f,p)) . (len (L_Cut (f,p))) by PARTFUN1:def_6
.= f . (len f) by A1, A2, JORDAN1B:4
.= f /. (len f) by A3, PARTFUN1:def_6 ;
::_thesis: verum
end;
theorem Th36: :: JORDAN1J:36
for f being non constant standard special_circular_sequence
for p, q being Point of (TOP-REAL 2)
for g being connected Subset of (TOP-REAL 2) st p in RightComp f & q in LeftComp f & p in g & q in g holds
g meets L~ f
proof
let f be non constant standard special_circular_sequence; ::_thesis: for p, q being Point of (TOP-REAL 2)
for g being connected Subset of (TOP-REAL 2) st p in RightComp f & q in LeftComp f & p in g & q in g holds
g meets L~ f
let p, q be Point of (TOP-REAL 2); ::_thesis: for g being connected Subset of (TOP-REAL 2) st p in RightComp f & q in LeftComp f & p in g & q in g holds
g meets L~ f
let g be connected Subset of (TOP-REAL 2); ::_thesis: ( p in RightComp f & q in LeftComp f & p in g & q in g implies g meets L~ f )
assume that
A1: p in RightComp f and
A2: q in LeftComp f and
A3: p in g and
A4: q in g ; ::_thesis: g meets L~ f
assume g misses L~ f ; ::_thesis: contradiction
then g c= (L~ f) ` by TDLAT_1:2;
then reconsider A = g as Subset of ((TOP-REAL 2) | ((L~ f) `)) by PRE_TOPC:8;
RightComp f is_a_component_of (L~ f) ` by GOBOARD9:def_2;
then consider R being Subset of ((TOP-REAL 2) | ((L~ f) `)) such that
A5: R = RightComp f and
A6: R is a_component by CONNSP_1:def_6;
R /\ A <> {} by A1, A3, A5, XBOOLE_0:def_4;
then A7: R meets A by XBOOLE_0:def_7;
LeftComp f is_a_component_of (L~ f) ` by GOBOARD9:def_1;
then consider L being Subset of ((TOP-REAL 2) | ((L~ f) `)) such that
A8: L = LeftComp f and
A9: L is a_component by CONNSP_1:def_6;
L /\ A <> {} by A2, A4, A8, XBOOLE_0:def_4;
then A10: L meets A by XBOOLE_0:def_7;
A is connected by CONNSP_1:23;
hence contradiction by A5, A6, A8, A9, A7, A10, JORDAN2C:92, SPRECT_4:6; ::_thesis: verum
end;
registration
cluster non empty non trivial V13() V16( NAT ) V17( the U1 of (TOP-REAL 2)) Function-like one-to-one non constant V26() FinSequence-like FinSubsequence-like special unfolded s.n.c. being_S-Seq s.c.c. standard for FinSequence of the U1 of (TOP-REAL 2);
existence
ex b1 being being_S-Seq FinSequence of (TOP-REAL 2) st
( not b1 is constant & b1 is standard & b1 is s.c.c. )
proof
set n = the Element of NAT ;
set C = the Simple_closed_curve;
A1: Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is_sequence_on Gauge ( the Simple_closed_curve, the Element of NAT ) by JORDAN1G:4;
take Upper_Seq ( the Simple_closed_curve, the Element of NAT ) ; ::_thesis: ( not Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is constant & Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is standard & Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is s.c.c. )
len (Upper_Seq ( the Simple_closed_curve, the Element of NAT )) >= 2 by TOPREAL1:def_8;
hence ( not Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is constant & Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is standard & Upper_Seq ( the Simple_closed_curve, the Element of NAT ) is s.c.c. ) by A1, JGRAPH_1:12, JORDAN8:5; ::_thesis: verum
end;
end;
theorem Th37: :: JORDAN1J:37
for f being S-Sequence_in_R2
for p being Point of (TOP-REAL 2) st p in rng f holds
L_Cut (f,p) = mid (f,(p .. f),(len f))
proof
let f be S-Sequence_in_R2; ::_thesis: for p being Point of (TOP-REAL 2) st p in rng f holds
L_Cut (f,p) = mid (f,(p .. f),(len f))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in rng f implies L_Cut (f,p) = mid (f,(p .. f),(len f)) )
A1: len f >= 2 by TOPREAL1:def_8;
assume p in rng f ; ::_thesis: L_Cut (f,p) = mid (f,(p .. f),(len f))
then consider i being Nat such that
A2: i in dom f and
A3: f . i = p by FINSEQ_2:10;
A4: 0 + 1 <= i by A2, FINSEQ_3:25;
A5: i <= len f by A2, FINSEQ_3:25;
percases ( i > 1 or i = 1 ) by A4, XXREAL_0:1;
suppose i > 1 ; ::_thesis: L_Cut (f,p) = mid (f,(p .. f),(len f))
then A6: (Index (p,f)) + 1 = i by A3, A5, JORDAN3:12;
then L_Cut (f,p) = mid (f,((Index (p,f)) + 1),(len f)) by A3, JORDAN3:def_3;
hence L_Cut (f,p) = mid (f,(p .. f),(len f)) by A2, A3, A6, FINSEQ_5:11; ::_thesis: verum
end;
supposeA7: i = 1 ; ::_thesis: L_Cut (f,p) = mid (f,(p .. f),(len f))
thus L_Cut (f,p) = L_Cut (f,(f /. i)) by A2, A3, PARTFUN1:def_6
.= f by A7, JORDAN5B:27
.= mid (f,1,(len f)) by A1, FINSEQ_6:120, XXREAL_0:2
.= mid (f,(p .. f),(len f)) by A2, A3, A7, FINSEQ_5:11 ; ::_thesis: verum
end;
end;
end;
theorem Th38: :: JORDAN1J:38
for M being Go-board
for f being S-Sequence_in_R2 st f is_sequence_on M holds
for p being Point of (TOP-REAL 2) st p in rng f holds
R_Cut (f,p) is_sequence_on M
proof
let M be Go-board; ::_thesis: for f being S-Sequence_in_R2 st f is_sequence_on M holds
for p being Point of (TOP-REAL 2) st p in rng f holds
R_Cut (f,p) is_sequence_on M
let f be S-Sequence_in_R2; ::_thesis: ( f is_sequence_on M implies for p being Point of (TOP-REAL 2) st p in rng f holds
R_Cut (f,p) is_sequence_on M )
assume A1: f is_sequence_on M ; ::_thesis: for p being Point of (TOP-REAL 2) st p in rng f holds
R_Cut (f,p) is_sequence_on M
let p be Point of (TOP-REAL 2); ::_thesis: ( p in rng f implies R_Cut (f,p) is_sequence_on M )
assume p in rng f ; ::_thesis: R_Cut (f,p) is_sequence_on M
then R_Cut (f,p) = mid (f,1,(p .. f)) by JORDAN1G:49;
hence R_Cut (f,p) is_sequence_on M by A1, JORDAN1H:27; ::_thesis: verum
end;
theorem Th39: :: JORDAN1J:39
for M being Go-board
for f being S-Sequence_in_R2 st f is_sequence_on M holds
for p being Point of (TOP-REAL 2) st p in rng f holds
L_Cut (f,p) is_sequence_on M
proof
let M be Go-board; ::_thesis: for f being S-Sequence_in_R2 st f is_sequence_on M holds
for p being Point of (TOP-REAL 2) st p in rng f holds
L_Cut (f,p) is_sequence_on M
let f be S-Sequence_in_R2; ::_thesis: ( f is_sequence_on M implies for p being Point of (TOP-REAL 2) st p in rng f holds
L_Cut (f,p) is_sequence_on M )
assume A1: f is_sequence_on M ; ::_thesis: for p being Point of (TOP-REAL 2) st p in rng f holds
L_Cut (f,p) is_sequence_on M
let p be Point of (TOP-REAL 2); ::_thesis: ( p in rng f implies L_Cut (f,p) is_sequence_on M )
assume p in rng f ; ::_thesis: L_Cut (f,p) is_sequence_on M
then L_Cut (f,p) = mid (f,(p .. f),(len f)) by Th37;
hence L_Cut (f,p) is_sequence_on M by A1, JORDAN1H:27; ::_thesis: verum
end;
theorem Th40: :: JORDAN1J:40
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G holds
for i, j being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f holds
G * (i,j) in rng f
proof
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G holds
for i, j being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f holds
G * (i,j) in rng f
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G implies for i, j being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f holds
G * (i,j) in rng f )
assume A1: f is_sequence_on G ; ::_thesis: for i, j being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f holds
G * (i,j) in rng f
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= len G & 1 <= j & j <= width G & G * (i,j) in L~ f implies G * (i,j) in rng f )
assume that
A2: 1 <= i and
A3: i <= len G and
A4: 1 <= j and
A5: j <= width G ; ::_thesis: ( not G * (i,j) in L~ f or G * (i,j) in rng f )
assume G * (i,j) in L~ f ; ::_thesis: G * (i,j) in rng f
then consider k being Element of NAT such that
A6: 1 <= k and
A7: k + 1 <= len f and
A8: G * (i,j) in LSeg ((f /. k),(f /. (k + 1))) by SPPOL_2:14;
consider i1, j1, i2, j2 being Element of NAT such that
A9: [i1,j1] in Indices G and
A10: f /. k = G * (i1,j1) and
A11: [i2,j2] in Indices G and
A12: f /. (k + 1) = G * (i2,j2) and
A13: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A6, A7, JORDAN8:3;
A14: 1 <= i1 by A9, MATRIX_1:38;
A15: 1 <= j2 by A11, MATRIX_1:38;
A16: i2 <= len G by A11, MATRIX_1:38;
k + 1 >= 1 by NAT_1:11;
then A17: k + 1 in dom f by A7, FINSEQ_3:25;
A18: 1 <= j1 by A9, MATRIX_1:38;
k < len f by A7, NAT_1:13;
then A19: k in dom f by A6, FINSEQ_3:25;
A20: i1 <= len G by A9, MATRIX_1:38;
A21: j2 <= width G by A11, MATRIX_1:38;
A22: 1 <= i2 by A11, MATRIX_1:38;
A23: j1 <= width G by A9, MATRIX_1:38;
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A13;
supposeA24: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: G * (i,j) in rng f
j1 <= j1 + 1 by NAT_1:11;
then A25: (G * (i1,j1)) `2 <= (G * (i1,(j1 + 1))) `2 by A14, A20, A18, A21, A24, JORDAN1A:19;
then (G * (i1,j1)) `2 <= (G * (i,j)) `2 by A8, A10, A12, A24, TOPREAL1:4;
then A26: j1 <= j by A2, A3, A4, A14, A20, A23, Th2;
A27: (G * (i1,j1)) `1 <= (G * (i1,(j1 + 1))) `1 by A14, A20, A18, A23, A15, A21, A24, JORDAN1A:18;
then (G * (i1,j1)) `1 <= (G * (i,j)) `1 by A8, A10, A12, A24, TOPREAL1:3;
then A28: i1 <= i by A2, A4, A5, A20, A18, A23, Th1;
(G * (i,j)) `2 <= (G * (i1,(j1 + 1))) `2 by A8, A10, A12, A24, A25, TOPREAL1:4;
then j <= j1 + 1 by A2, A3, A5, A14, A20, A15, A24, Th2;
then A29: ( j = j1 or j = j1 + 1 ) by A26, NAT_1:9;
(G * (i,j)) `1 <= (G * (i1,(j1 + 1))) `1 by A8, A10, A12, A24, A27, TOPREAL1:3;
then i <= i1 by A3, A4, A5, A14, A15, A21, A24, Th1;
then i = i1 by A28, XXREAL_0:1;
hence G * (i,j) in rng f by A10, A12, A19, A17, A24, A29, PARTFUN2:2; ::_thesis: verum
end;
supposeA30: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: G * (i,j) in rng f
i1 <= i1 + 1 by NAT_1:11;
then A31: (G * (i1,j1)) `1 <= (G * ((i1 + 1),j1)) `1 by A14, A18, A23, A16, A30, JORDAN1A:18;
then (G * (i1,j1)) `1 <= (G * (i,j)) `1 by A8, A10, A12, A30, TOPREAL1:3;
then A32: i1 <= i by A2, A4, A5, A20, A18, A23, Th1;
A33: (G * (i1,j1)) `2 <= (G * ((i1 + 1),j1)) `2 by A14, A20, A18, A23, A22, A16, A30, JORDAN1A:19;
then (G * (i1,j1)) `2 <= (G * (i,j)) `2 by A8, A10, A12, A30, TOPREAL1:4;
then A34: j1 <= j by A2, A3, A4, A14, A20, A23, Th2;
(G * (i,j)) `1 <= (G * ((i1 + 1),j1)) `1 by A8, A10, A12, A30, A31, TOPREAL1:3;
then i <= i1 + 1 by A3, A4, A5, A18, A23, A22, A30, Th1;
then A35: ( i = i1 or i = i1 + 1 ) by A32, NAT_1:9;
(G * (i,j)) `2 <= (G * ((i1 + 1),j1)) `2 by A8, A10, A12, A30, A33, TOPREAL1:4;
then j <= j1 by A2, A3, A5, A18, A22, A16, A30, Th2;
then j = j1 by A34, XXREAL_0:1;
hence G * (i,j) in rng f by A10, A12, A19, A17, A30, A35, PARTFUN2:2; ::_thesis: verum
end;
supposeA36: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: G * (i,j) in rng f
i2 <= i2 + 1 by NAT_1:11;
then A37: (G * (i2,j1)) `1 <= (G * ((i2 + 1),j1)) `1 by A20, A18, A23, A22, A36, JORDAN1A:18;
then (G * (i2,j1)) `1 <= (G * (i,j)) `1 by A8, A10, A12, A36, TOPREAL1:3;
then A38: i2 <= i by A2, A4, A5, A18, A23, A16, Th1;
A39: (G * (i2,j1)) `2 <= (G * ((i2 + 1),j1)) `2 by A14, A20, A18, A23, A22, A16, A36, JORDAN1A:19;
then (G * (i2,j1)) `2 <= (G * (i,j)) `2 by A8, A10, A12, A36, TOPREAL1:4;
then A40: j1 <= j by A2, A3, A4, A23, A22, A16, Th2;
(G * (i,j)) `1 <= (G * ((i2 + 1),j1)) `1 by A8, A10, A12, A36, A37, TOPREAL1:3;
then i <= i2 + 1 by A3, A4, A5, A14, A18, A23, A36, Th1;
then A41: ( i = i2 or i = i2 + 1 ) by A38, NAT_1:9;
(G * (i,j)) `2 <= (G * ((i2 + 1),j1)) `2 by A8, A10, A12, A36, A39, TOPREAL1:4;
then j <= j1 by A2, A3, A5, A14, A20, A18, A36, Th2;
then j = j1 by A40, XXREAL_0:1;
hence G * (i,j) in rng f by A10, A12, A19, A17, A36, A41, PARTFUN2:2; ::_thesis: verum
end;
supposeA42: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: G * (i,j) in rng f
j2 <= j2 + 1 by NAT_1:11;
then A43: (G * (i1,j2)) `2 <= (G * (i1,(j2 + 1))) `2 by A14, A20, A23, A15, A42, JORDAN1A:19;
then (G * (i1,j2)) `2 <= (G * (i,j)) `2 by A8, A10, A12, A42, TOPREAL1:4;
then A44: j2 <= j by A2, A3, A4, A14, A20, A21, Th2;
A45: (G * (i1,j2)) `1 <= (G * (i1,(j2 + 1))) `1 by A14, A20, A18, A23, A15, A21, A42, JORDAN1A:18;
then (G * (i1,j2)) `1 <= (G * (i,j)) `1 by A8, A10, A12, A42, TOPREAL1:3;
then A46: i1 <= i by A2, A4, A5, A20, A15, A21, Th1;
(G * (i,j)) `2 <= (G * (i1,(j2 + 1))) `2 by A8, A10, A12, A42, A43, TOPREAL1:4;
then j <= j2 + 1 by A2, A3, A5, A14, A20, A18, A42, Th2;
then A47: ( j = j2 or j = j2 + 1 ) by A44, NAT_1:9;
(G * (i,j)) `1 <= (G * (i1,(j2 + 1))) `1 by A8, A10, A12, A42, A45, TOPREAL1:3;
then i <= i1 by A3, A4, A5, A14, A18, A23, A42, Th1;
then i = i1 by A46, XXREAL_0:1;
hence G * (i,j) in rng f by A10, A12, A19, A17, A42, A47, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:41
for f being S-Sequence_in_R2
for g being FinSequence of (TOP-REAL 2) st g is unfolded & g is s.n.c. & g is one-to-one & (L~ f) /\ (L~ g) = {(f /. 1)} & f /. 1 = g /. (len g) & ( for i being Element of NAT st 1 <= i & i + 2 <= len f holds
(LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) & ( for i being Element of NAT st 2 <= i & i + 1 <= len g holds
(LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) holds
f ^ g is s.c.c.
proof
let f be S-Sequence_in_R2; ::_thesis: for g being FinSequence of (TOP-REAL 2) st g is unfolded & g is s.n.c. & g is one-to-one & (L~ f) /\ (L~ g) = {(f /. 1)} & f /. 1 = g /. (len g) & ( for i being Element of NAT st 1 <= i & i + 2 <= len f holds
(LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) & ( for i being Element of NAT st 2 <= i & i + 1 <= len g holds
(LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) holds
f ^ g is s.c.c.
let g be FinSequence of (TOP-REAL 2); ::_thesis: ( g is unfolded & g is s.n.c. & g is one-to-one & (L~ f) /\ (L~ g) = {(f /. 1)} & f /. 1 = g /. (len g) & ( for i being Element of NAT st 1 <= i & i + 2 <= len f holds
(LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) & ( for i being Element of NAT st 2 <= i & i + 1 <= len g holds
(LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ) implies f ^ g is s.c.c. )
assume that
A1: ( g is unfolded & g is s.n.c. & g is one-to-one ) and
A2: (L~ f) /\ (L~ g) = {(f /. 1)} and
A3: f /. 1 = g /. (len g) and
A4: for i being Element of NAT st 1 <= i & i + 2 <= len f holds
(LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} and
A5: for i being Element of NAT st 2 <= i & i + 1 <= len g holds
(LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} ; ::_thesis: f ^ g is s.c.c.
let i, j be Element of NAT ; :: according to GOBOARD5:def_4 ::_thesis: ( j <= i + 1 or ( ( i <= 1 or len (f ^ g) <= j ) & len (f ^ g) <= j + 1 ) or LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) )
assume that
A6: i + 1 < j and
A7: ( ( i > 1 & j < len (f ^ g) ) or j + 1 < len (f ^ g) ) ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
A8: j + 1 <= len (f ^ g) by A7, NAT_1:13;
A9: now__::_thesis:_for_i_being_Element_of_NAT_st_2_<=_i_&_i_+_1_<=_len_g_holds_
LSeg_(g,i)_misses_LSeg_((f_/._(len_f)),(g_/._1))
let i be Element of NAT ; ::_thesis: ( 2 <= i & i + 1 <= len g implies LSeg (g,i) misses LSeg ((f /. (len f)),(g /. 1)) )
assume that
A10: 2 <= i and
A11: i + 1 <= len g ; ::_thesis: LSeg (g,i) misses LSeg ((f /. (len f)),(g /. 1))
(LSeg (g,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} by A5, A10, A11;
hence LSeg (g,i) misses LSeg ((f /. (len f)),(g /. 1)) by XBOOLE_0:def_7; ::_thesis: verum
end;
A12: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_+_2_<=_len_f_holds_
LSeg_(f,i)_misses_LSeg_((f_/._(len_f)),(g_/._1))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 2 <= len f implies LSeg (f,i) misses LSeg ((f /. (len f)),(g /. 1)) )
assume that
A13: 1 <= i and
A14: i + 2 <= len f ; ::_thesis: LSeg (f,i) misses LSeg ((f /. (len f)),(g /. 1))
(LSeg (f,i)) /\ (LSeg ((f /. (len f)),(g /. 1))) = {} by A4, A13, A14;
hence LSeg (f,i) misses LSeg ((f /. (len f)),(g /. 1)) by XBOOLE_0:def_7; ::_thesis: verum
end;
percases ( i = 0 or i <> 0 ) ;
suppose i = 0 ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
then LSeg ((f ^ g),i) = {} by TOPREAL1:def_3;
then (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {} ;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA15: i <> 0 ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
A16: len (f ^ g) = (len f) + (len g) by FINSEQ_1:22;
i <= i + 1 by NAT_1:11;
then A17: i < j by A6, XXREAL_0:2;
now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j)
percases ( j + 1 <= len f or j + 1 > len f ) ;
supposeA18: j + 1 <= len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
j <= j + 1 by NAT_1:11;
then i < j + 1 by A17, XXREAL_0:2;
then i < len f by A18, XXREAL_0:2;
then i + 1 <= len f by NAT_1:13;
then A19: LSeg ((f ^ g),i) = LSeg (f,i) by SPPOL_2:6;
LSeg ((f ^ g),j) = LSeg (f,j) by A18, SPPOL_2:6;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A6, A19, TOPREAL1:def_7; ::_thesis: verum
end;
suppose j + 1 > len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
then A20: len f <= j by NAT_1:13;
then reconsider j9 = j - (len f) as Element of NAT by INT_1:5;
A21: (j + 1) - (len f) <= len g by A8, A16, XREAL_1:20;
then A22: j9 + 1 <= len g ;
A23: (len f) + j9 = j ;
now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j)
percases ( i <= len f or i > len f ) ;
supposeA24: i <= len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j)
percases ( i = len f or i <> len f ) ;
supposeA25: i = len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
not g is empty by A22;
then A26: LSeg ((f ^ g),i) = LSeg ((f /. (len f)),(g /. 1)) by A25, SPPOL_2:8;
((len f) + 1) + 1 <= j by A6, A25, NAT_1:13;
then (len f) + (1 + 1) <= j ;
then A27: 1 + 1 <= j9 by XREAL_1:19;
then LSeg ((f ^ g),j) = LSeg (g,j9) by A23, SPPOL_2:7, XXREAL_0:2;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A9, A22, A27, A26; ::_thesis: verum
end;
suppose i <> len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
then i < len f by A24, XXREAL_0:1;
then i + 1 <= len f by NAT_1:13;
then A28: LSeg ((f ^ g),i) = LSeg (f,i) by SPPOL_2:6;
now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j)
percases ( j = len f or j <> len f ) ;
supposeA29: j = len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
then (i + 1) + 1 <= len f by A6, NAT_1:13;
then A30: i + (1 + 1) <= len f ;
not g is empty by A8, A16, A29, XREAL_1:6;
then LSeg ((f ^ g),j) = LSeg ((f /. (len f)),(g /. 1)) by A29, SPPOL_2:8;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A12, A15, A28, A30, NAT_1:14; ::_thesis: verum
end;
supposeA31: j <> len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
A32: len f >= 2 by TOPREAL1:def_8;
A33: LSeg ((f ^ g),i) c= L~ f by A28, TOPREAL3:19;
len f < j by A20, A31, XXREAL_0:1;
then (len f) + 1 <= j by NAT_1:13;
then A34: 1 <= j9 by XREAL_1:19;
then A35: LSeg ((f ^ g),((len f) + j9)) = LSeg (g,j9) by SPPOL_2:7;
then LSeg ((f ^ g),j) c= L~ g by TOPREAL3:19;
then A36: (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) c= {(f /. 1)} by A2, A33, XBOOLE_1:27;
now__::_thesis:_LSeg_((f_^_g),i)_misses_LSeg_((f_^_g),j)
percases ( (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {} or (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {(f /. 1)} ) by A36, ZFMISC_1:33;
suppose (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {} ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by XBOOLE_0:def_7; ::_thesis: verum
end;
suppose (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) = {(f /. 1)} ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
then A37: f /. 1 in (LSeg ((f ^ g),i)) /\ (LSeg ((f ^ g),j)) by TARSKI:def_1;
then A38: f /. 1 in LSeg ((f ^ g),i) by XBOOLE_0:def_4;
j9 < len g by A22, NAT_1:13;
then A39: j9 in dom g by A34, FINSEQ_3:25;
j9 + 1 >= 1 by NAT_1:11;
then A40: j9 + 1 in dom g by A21, FINSEQ_3:25;
f /. 1 in LSeg ((f ^ g),j) by A37, XBOOLE_0:def_4;
then j9 + 1 = len g by A1, A3, A35, A39, A40, GOBOARD2:2;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A7, A16, A28, A32, A38, JORDAN5B:30; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum
end;
supposeA41: i > len f ; ::_thesis: LSeg ((f ^ g),i) misses LSeg ((f ^ g),j)
then j <> len f by A6, NAT_1:13;
then len f < j by A20, XXREAL_0:1;
then (len f) + 1 <= j by NAT_1:13;
then 1 <= j9 by XREAL_1:19;
then A42: LSeg ((f ^ g),((len f) + j9)) = LSeg (g,j9) by SPPOL_2:7;
reconsider i9 = i - (len f) as Element of NAT by A41, INT_1:5;
(len f) + 1 <= i by A41, NAT_1:13;
then 1 <= i9 by XREAL_1:19;
then A43: LSeg ((f ^ g),((len f) + i9)) = LSeg (g,i9) by SPPOL_2:7;
(i + 1) - (len f) < j9 by A6, XREAL_1:9;
then i9 + 1 < j9 ;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) by A1, A43, A42, TOPREAL1:def_7; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^ g),i) misses LSeg ((f ^ g),j) ; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1J:42
for n being Element of NAT
for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) )
proof
let n be Element of NAT ; ::_thesis: for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) )
let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) )
set G = Gauge (C,n);
defpred S1[ Nat] means ( 1 <= $1 & $1 < len (Gauge (C,n)) & ((Gauge (C,n)) * (2,$1)) `2 < (W-min C) `2 );
A1: for k being Nat st S1[k] holds
k <= len (Gauge (C,n)) ;
A2: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
(SW-corner C) `2 <= (W-min C) `2 by PSCOMP_1:30;
then A3: S-bound C <= (W-min C) `2 by EUCLID:52;
A4: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A5: 1 < len (Gauge (C,n)) by XXREAL_0:2;
A6: 2 <= len (Gauge (C,n)) by A4, XXREAL_0:2;
then ((Gauge (C,n)) * (2,2)) `2 = S-bound C by JORDAN8:13;
then ((Gauge (C,n)) * (2,1)) `2 < S-bound C by A2, A6, GOBOARD5:4;
then ((Gauge (C,n)) * (2,1)) `2 < (W-min C) `2 by A3, XXREAL_0:2;
then A7: ex k being Nat st S1[k] by A5;
ex i being Nat st
( S1[i] & ( for n being Nat st S1[n] holds
n <= i ) ) from NAT_1:sch_6(A1, A7);
then consider i being Nat such that
A8: 1 <= i and
A9: i < len (Gauge (C,n)) and
A10: ((Gauge (C,n)) * (2,i)) `2 < (W-min C) `2 and
A11: for n being Nat st S1[n] holds
n <= i ;
reconsider i = i as Element of NAT by ORDINAL1:def_12;
A12: (W-min C) `1 = W-bound C by EUCLID:52;
then A13: ((Gauge (C,n)) * (2,i)) `1 = (W-min C) `1 by A8, A9, JORDAN8:11;
A14: i + 1 <= len (Gauge (C,n)) by A9, NAT_1:13;
then A15: (W-min C) `1 = ((Gauge (C,n)) * (2,(i + 1))) `1 by A12, JORDAN8:11, NAT_1:12;
A16: i < i + 1 by NAT_1:13;
A17: 1 <= i + 1 by NAT_1:12;
now__::_thesis:_not_i_+_1_=_len_(Gauge_(C,n))
assume i + 1 = len (Gauge (C,n)) ; ::_thesis: contradiction
then (len (Gauge (C,n))) -' 1 = i by NAT_D:34;
then A18: ((Gauge (C,n)) * (2,i)) `2 = N-bound C by A6, JORDAN8:14;
(NW-corner C) `2 >= (W-min C) `2 by PSCOMP_1:30;
hence contradiction by A10, A18, EUCLID:52; ::_thesis: verum
end;
then i + 1 < len (Gauge (C,n)) by A14, XXREAL_0:1;
then (W-min C) `2 <= ((Gauge (C,n)) * (2,(i + 1))) `2 by A11, A17, A16;
then A19: W-min C in LSeg (((Gauge (C,n)) * (2,i)),((Gauge (C,n)) * (2,(i + 1)))) by A10, A13, A15, GOBOARD7:7;
take i ; ::_thesis: ( 1 <= i & i + 1 <= len (Gauge (C,n)) & W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) )
thus ( 1 <= i & i + 1 <= len (Gauge (C,n)) ) by A8, A9, NAT_1:13; ::_thesis: ( W-min C in cell ((Gauge (C,n)),1,i) & W-min C <> (Gauge (C,n)) * (2,i) )
len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then LSeg (((Gauge (C,n)) * ((1 + 1),i)),((Gauge (C,n)) * ((1 + 1),(i + 1)))) c= cell ((Gauge (C,n)),1,i) by A5, A8, A9, GOBOARD5:18;
hence W-min C in cell ((Gauge (C,n)),1,i) by A19; ::_thesis: W-min C <> (Gauge (C,n)) * (2,i)
thus W-min C <> (Gauge (C,n)) * (2,i) by A10; ::_thesis: verum
end;
theorem Th43: :: JORDAN1J:43
for f being S-Sequence_in_R2
for p being Point of (TOP-REAL 2) st p in L~ f & f . (len f) in L~ (R_Cut (f,p)) holds
f . (len f) = p
proof
let f be S-Sequence_in_R2; ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f & f . (len f) in L~ (R_Cut (f,p)) holds
f . (len f) = p
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f & f . (len f) in L~ (R_Cut (f,p)) implies f . (len f) = p )
assume that
A1: p in L~ f and
A2: f . (len f) in L~ (R_Cut (f,p)) ; ::_thesis: f . (len f) = p
A3: L~ f = L~ (Rev f) by SPPOL_2:22;
A4: (Rev f) . 1 = f . (len f) by FINSEQ_5:62;
L_Cut ((Rev f),p) = Rev (R_Cut (f,p)) by A1, JORDAN3:22;
then (Rev f) . 1 in L~ (L_Cut ((Rev f),p)) by A2, A4, SPPOL_2:22;
hence f . (len f) = p by A1, A3, A4, JORDAN1E:7; ::_thesis: verum
end;
theorem Th44: :: JORDAN1J:44
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) holds R_Cut (f,p) <> {}
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) holds R_Cut (f,p) <> {}
let p be Point of (TOP-REAL 2); ::_thesis: R_Cut (f,p) <> {}
percases ( p <> f . 1 or p = f . 1 ) ;
suppose p <> f . 1 ; ::_thesis: R_Cut (f,p) <> {}
then R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> by JORDAN3:def_4;
hence R_Cut (f,p) <> {} ; ::_thesis: verum
end;
suppose p = f . 1 ; ::_thesis: R_Cut (f,p) <> {}
then R_Cut (f,p) = <*p*> by JORDAN3:def_4;
hence R_Cut (f,p) <> {} ; ::_thesis: verum
end;
end;
end;
theorem Th45: :: JORDAN1J:45
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
(R_Cut (f,p)) /. (len (R_Cut (f,p))) = p
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
(R_Cut (f,p)) /. (len (R_Cut (f,p))) = p
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies (R_Cut (f,p)) /. (len (R_Cut (f,p))) = p )
assume A1: p in L~ f ; ::_thesis: (R_Cut (f,p)) /. (len (R_Cut (f,p))) = p
not R_Cut (f,p) is empty by Th44;
then len (R_Cut (f,p)) in dom (R_Cut (f,p)) by FINSEQ_5:6;
hence (R_Cut (f,p)) /. (len (R_Cut (f,p))) = (R_Cut (f,p)) . (len (R_Cut (f,p))) by PARTFUN1:def_6
.= p by A1, JORDAN3:24 ;
::_thesis: verum
end;
theorem Th46: :: JORDAN1J:46
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ (Upper_Seq (C,n)) & p `1 = E-bound (L~ (Cage (C,n))) holds
p = E-max (L~ (Cage (C,n)))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ (Upper_Seq (C,n)) & p `1 = E-bound (L~ (Cage (C,n))) holds
p = E-max (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ (Upper_Seq (C,n)) & p `1 = E-bound (L~ (Cage (C,n))) holds
p = E-max (L~ (Cage (C,n)))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ (Upper_Seq (C,n)) & p `1 = E-bound (L~ (Cage (C,n))) implies p = E-max (L~ (Cage (C,n))) )
set Ca = Cage (C,n);
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
set Wmin = W-min (L~ (Cage (C,n)));
set Smax = S-max (L~ (Cage (C,n)));
set Smin = S-min (L~ (Cage (C,n)));
set Emin = E-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Nbo = N-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Sbo = S-bound (L~ (Cage (C,n)));
set NE = NE-corner (L~ (Cage (C,n)));
assume that
A1: p in L~ (Upper_Seq (C,n)) and
A2: p `1 = E-bound (L~ (Cage (C,n))) and
A3: p <> E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
A4: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
1 in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A5: (Upper_Seq (C,n)) . 1 = W-min (L~ (Cage (C,n))) by A4, PARTFUN1:def_6;
W-bound (L~ (Cage (C,n))) <> E-bound (L~ (Cage (C,n))) by SPRECT_1:31;
then p <> (Upper_Seq (C,n)) . 1 by A2, A5, EUCLID:52;
then reconsider g = R_Cut ((Upper_Seq (C,n)),p) as being_S-Seq FinSequence of (TOP-REAL 2) by A1, JORDAN3:35;
<*p*> is_in_the_area_of Cage (C,n) by A1, JORDAN1E:17, SPRECT_3:46;
then A6: g is_in_the_area_of Cage (C,n) by A1, JORDAN1E:17, SPRECT_3:52;
len g in dom g by FINSEQ_5:6;
then A7: g /. (len g) = g . (len g) by PARTFUN1:def_6
.= p by A1, JORDAN3:24 ;
(g /. 1) `1 = ((Upper_Seq (C,n)) /. 1) `1 by A1, SPRECT_3:22
.= (W-min (L~ (Cage (C,n)))) `1 by JORDAN1F:5
.= W-bound (L~ (Cage (C,n))) by EUCLID:52 ;
then A8: g is_a_h.c._for Cage (C,n) by A2, A6, A7, SPRECT_2:def_2;
A9: (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) by JORDAN1F:6;
1 in dom (Lower_Seq (C,n)) by FINSEQ_5:6;
then A10: (Lower_Seq (C,n)) . 1 = E-max (L~ (Cage (C,n))) by A9, PARTFUN1:def_6;
len (Cage (C,n)) > 4 by GOBOARD7:34;
then A11: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2;
now__::_thesis:_contradiction
percases ( E-max (L~ (Cage (C,n))) <> NE-corner (L~ (Cage (C,n))) or E-max (L~ (Cage (C,n))) = NE-corner (L~ (Cage (C,n))) ) ;
supposeA12: E-max (L~ (Cage (C,n))) <> NE-corner (L~ (Cage (C,n))) ; ::_thesis: contradiction
A13: not NE-corner (L~ (Cage (C,n))) in rng (Cage (C,n))
proof
A14: (NE-corner (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
A15: (NE-corner (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
then (NE-corner (L~ (Cage (C,n)))) `2 >= S-bound (L~ (Cage (C,n))) by SPRECT_1:22;
then NE-corner (L~ (Cage (C,n))) in { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = E-bound (L~ (Cage (C,n))) & p1 `2 <= N-bound (L~ (Cage (C,n))) & p1 `2 >= S-bound (L~ (Cage (C,n))) ) } by A14, A15;
then A16: NE-corner (L~ (Cage (C,n))) in LSeg ((SE-corner (L~ (Cage (C,n)))),(NE-corner (L~ (Cage (C,n))))) by SPRECT_1:23;
assume NE-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) ; ::_thesis: contradiction
then NE-corner (L~ (Cage (C,n))) in (LSeg ((SE-corner (L~ (Cage (C,n)))),(NE-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A11, A16, XBOOLE_0:def_4;
then A17: (NE-corner (L~ (Cage (C,n)))) `2 <= (E-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:47;
A18: (E-max (L~ (Cage (C,n)))) `1 = (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:45;
(E-max (L~ (Cage (C,n)))) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:46;
then (E-max (L~ (Cage (C,n)))) `2 = (NE-corner (L~ (Cage (C,n)))) `2 by A17, XXREAL_0:1;
hence contradiction by A12, A18, TOPREAL3:6; ::_thesis: verum
end;
S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by Th12;
then R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) = mid ((Lower_Seq (C,n)),1,((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)))) by JORDAN1G:49;
then A19: rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) c= rng (Lower_Seq (C,n)) by FINSEQ_6:119;
rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by JORDAN1G:39;
then rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) c= rng (Cage (C,n)) by A19, XBOOLE_1:1;
then not NE-corner (L~ (Cage (C,n))) in rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A13;
then rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) misses {(NE-corner (L~ (Cage (C,n))))} by ZFMISC_1:50;
then rng (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) misses rng <*(NE-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38;
then A20: rng (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) misses rng <*(NE-corner (L~ (Cage (C,n))))*> by FINSEQ_5:57;
set h = (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>;
A21: <*(NE-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93;
A22: (((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. (len ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>))) `2 = (((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. ((len (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))))) + 1)) `2 by FINSEQ_2:16
.= (NE-corner (L~ (Cage (C,n)))) `2 by FINSEQ_4:67
.= N-bound (L~ (Cage (C,n))) by EUCLID:52 ;
E-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:14;
then A23: S-bound (L~ (Cage (C,n))) <= (E-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
A24: (Index ((S-max (L~ (Cage (C,n)))),(Lower_Seq (C,n)))) + 1 >= 0 + 1 by XREAL_1:7;
A25: S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by Th12;
then <*(S-max (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:46;
then R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) is_in_the_area_of Cage (C,n) by A25, JORDAN1E:18, SPRECT_3:52;
then A26: Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51;
(E-min (L~ (Cage (C,n)))) `2 < (E-max (L~ (Cage (C,n)))) `2 by SPRECT_2:53;
then A27: S-max (L~ (Cage (C,n))) <> (Lower_Seq (C,n)) . 1 by A10, A23, EUCLID:52;
then reconsider RCutLS = R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) as being_S-Seq FinSequence of (TOP-REAL 2) by A25, JORDAN3:35;
len ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = (len (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))))) + 1 by FINSEQ_2:16
.= (len (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) + 1 by FINSEQ_5:def_3
.= ((Index ((S-max (L~ (Cage (C,n)))),(Lower_Seq (C,n)))) + 1) + 1 by A25, A27, JORDAN3:25 ;
then A28: len ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) >= 1 + 1 by A24, XREAL_1:7;
A29: 2 <= len g by TOPREAL1:def_8;
1 in dom (Rev RCutLS) by FINSEQ_5:6;
then ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1 = (Rev RCutLS) /. 1 by FINSEQ_4:68
.= (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) /. (len (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) by FINSEQ_5:65
.= S-max (L~ (Cage (C,n))) by A25, Th45 ;
then A30: (((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
<*(NE-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:25;
then (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by A26, SPRECT_2:24;
then A31: (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is_a_v.c._for Cage (C,n) by A30, A22, SPRECT_2:def_3;
A32: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6;
A33: (Rev RCutLS) /. (len (Rev RCutLS)) = (Rev RCutLS) /. (len RCutLS) by FINSEQ_5:def_3
.= RCutLS /. 1 by FINSEQ_5:65
.= (Lower_Seq (C,n)) /. 1 by A25, SPRECT_3:22
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
then ((Rev RCutLS) /. (len (Rev RCutLS))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= (NE-corner (L~ (Cage (C,n)))) `1 by EUCLID:52
.= (<*(NE-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_4:16 ;
then ( (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is one-to-one & (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is special ) by A20, A21, FINSEQ_3:91, GOBOARD2:8;
then L~ g meets L~ ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by A8, A29, A28, A31, SPRECT_2:29;
then consider x being set such that
A34: x in L~ g and
A35: x in L~ ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A34;
A36: L~ ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = (L~ (Rev RCutLS)) \/ (LSeg (((Rev RCutLS) /. (len (Rev RCutLS))),(NE-corner (L~ (Cage (C,n)))))) by SPPOL_2:19;
A37: L~ RCutLS c= L~ (Lower_Seq (C,n)) by Th12, JORDAN3:41;
A38: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
A39: L~ g c= L~ (Upper_Seq (C,n)) by A1, JORDAN3:41;
then A40: x in L~ (Upper_Seq (C,n)) by A34;
now__::_thesis:_contradiction
percases ( x in L~ (Rev RCutLS) or x in LSeg (((Rev RCutLS) /. (len (Rev RCutLS))),(NE-corner (L~ (Cage (C,n))))) ) by A35, A36, XBOOLE_0:def_3;
suppose x in L~ (Rev RCutLS) ; ::_thesis: contradiction
then A41: x in L~ RCutLS by SPPOL_2:22;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A34, A39, A37, XBOOLE_0:def_4;
then A42: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
now__::_thesis:_contradiction
percases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A42, TARSKI:def_2;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A41, JORDAN1F:8;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A32, PARTFUN1:def_6;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by A25, Th43;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by A32, PARTFUN1:def_6;
then A43: W-min (L~ (Cage (C,n))) = S-max (L~ (Cage (C,n))) by JORDAN1F:8;
S-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:12;
then A44: W-bound (L~ (Cage (C,n))) <= (S-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:24;
(S-min (L~ (Cage (C,n)))) `1 < (S-max (L~ (Cage (C,n)))) `1 by SPRECT_2:55;
hence contradiction by A43, A44, EUCLID:52; ::_thesis: verum
end;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A34, JORDAN1F:7;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A38, PARTFUN1:def_6;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = p by A1, Th43;
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = p by A38, PARTFUN1:def_6;
hence contradiction by A3, JORDAN1F:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA45: x in LSeg (((Rev RCutLS) /. (len (Rev RCutLS))),(NE-corner (L~ (Cage (C,n))))) ; ::_thesis: contradiction
(E-max (L~ (Cage (C,n)))) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:46;
then A46: (E-max (L~ (Cage (C,n)))) `2 <= x `2 by A33, A45, TOPREAL1:4;
A47: (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
(NE-corner (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
then A48: x `1 = E-bound (L~ (Cage (C,n))) by A33, A45, A47, GOBOARD7:5;
L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
then x in E-most (L~ (Cage (C,n))) by A40, A48, SPRECT_2:13;
then x `2 <= (E-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:47;
then x `2 = (E-max (L~ (Cage (C,n)))) `2 by A46, XXREAL_0:1;
then x = E-max (L~ (Cage (C,n))) by A47, A48, TOPREAL3:6;
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A34, JORDAN1F:7;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A38, PARTFUN1:def_6;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = p by A1, Th43;
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = p by A38, PARTFUN1:def_6;
hence contradiction by A3, JORDAN1F:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA49: E-max (L~ (Cage (C,n))) = NE-corner (L~ (Cage (C,n))) ; ::_thesis: contradiction
E-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:14;
then A50: S-bound (L~ (Cage (C,n))) <= (E-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
set h = Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))));
A51: 2 <= len g by TOPREAL1:def_8;
A52: S-max (L~ (Cage (C,n))) in L~ (Lower_Seq (C,n)) by Th12;
then <*(S-max (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:46;
then R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) is_in_the_area_of Cage (C,n) by A52, JORDAN1E:18, SPRECT_3:52;
then A53: Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51;
(E-min (L~ (Cage (C,n)))) `2 < (E-max (L~ (Cage (C,n)))) `2 by SPRECT_2:53;
then S-max (L~ (Cage (C,n))) <> (Lower_Seq (C,n)) . 1 by A10, A50, EUCLID:52;
then reconsider RCutLS = R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))) as being_S-Seq FinSequence of (TOP-REAL 2) by A52, JORDAN3:35;
1 in dom (Rev RCutLS) by FINSEQ_5:6;
then (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) /. 1 = (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) /. (len (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) by FINSEQ_5:65
.= S-max (L~ (Cage (C,n))) by A52, Th45 ;
then A54: ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) /. 1) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
A55: Rev RCutLS is special ;
len RCutLS >= 2 by TOPREAL1:def_8;
then A56: len (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) >= 2 by FINSEQ_5:def_3;
(Rev RCutLS) /. (len (Rev RCutLS)) = (Rev RCutLS) /. (len RCutLS) by FINSEQ_5:def_3
.= RCutLS /. 1 by FINSEQ_5:65
.= (Lower_Seq (C,n)) /. 1 by A52, SPRECT_3:22
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
then ((Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) /. (len (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))))) `2 = N-bound (L~ (Cage (C,n))) by A49, EUCLID:52;
then Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) is_a_v.c._for Cage (C,n) by A53, A54, SPRECT_2:def_3;
then L~ g meets L~ (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) by A8, A55, A51, A56, SPRECT_2:29;
then consider x being set such that
A57: x in L~ g and
A58: x in L~ (Rev (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n))))))) by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A57;
A59: x in L~ RCutLS by A58, SPPOL_2:22;
A60: L~ g c= L~ (Upper_Seq (C,n)) by A1, JORDAN3:41;
A61: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
A62: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6;
L~ RCutLS c= L~ (Lower_Seq (C,n)) by Th12, JORDAN3:41;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A57, A60, A59, XBOOLE_0:def_4;
then A63: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
now__::_thesis:_contradiction
percases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A63, TARSKI:def_2;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A59, JORDAN1F:8;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),(S-max (L~ (Cage (C,n)))))) by A62, PARTFUN1:def_6;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by A52, Th43;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by A62, PARTFUN1:def_6;
then A64: W-min (L~ (Cage (C,n))) = S-max (L~ (Cage (C,n))) by JORDAN1F:8;
S-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:12;
then A65: W-bound (L~ (Cage (C,n))) <= (S-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:24;
(S-min (L~ (Cage (C,n)))) `1 < (S-max (L~ (Cage (C,n)))) `1 by SPRECT_2:55;
hence contradiction by A64, A65, EUCLID:52; ::_thesis: verum
end;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A57, JORDAN1F:7;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),p)) by A61, PARTFUN1:def_6;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = p by A1, Th43;
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = p by A61, PARTFUN1:def_6;
hence contradiction by A3, JORDAN1F:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
theorem :: JORDAN1J:47
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ (Lower_Seq (C,n)) & p `1 = W-bound (L~ (Cage (C,n))) holds
p = W-min (L~ (Cage (C,n)))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ (Lower_Seq (C,n)) & p `1 = W-bound (L~ (Cage (C,n))) holds
p = W-min (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ (Lower_Seq (C,n)) & p `1 = W-bound (L~ (Cage (C,n))) holds
p = W-min (L~ (Cage (C,n)))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ (Lower_Seq (C,n)) & p `1 = W-bound (L~ (Cage (C,n))) implies p = W-min (L~ (Cage (C,n))) )
set Ca = Cage (C,n);
set LS = Lower_Seq (C,n);
set US = Upper_Seq (C,n);
set Emax = E-max (L~ (Cage (C,n)));
set Nmin = N-min (L~ (Cage (C,n)));
set Nmax = N-max (L~ (Cage (C,n)));
set Wmax = W-max (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Sbo = S-bound (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Nbo = N-bound (L~ (Cage (C,n)));
set SW = SW-corner (L~ (Cage (C,n)));
assume that
A1: p in L~ (Lower_Seq (C,n)) and
A2: p `1 = W-bound (L~ (Cage (C,n))) and
A3: p <> W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
A4: (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) by JORDAN1F:6;
1 in dom (Lower_Seq (C,n)) by FINSEQ_5:6;
then A5: (Lower_Seq (C,n)) . 1 = E-max (L~ (Cage (C,n))) by A4, PARTFUN1:def_6;
E-bound (L~ (Cage (C,n))) <> W-bound (L~ (Cage (C,n))) by SPRECT_1:31;
then p <> (Lower_Seq (C,n)) . 1 by A2, A5, EUCLID:52;
then reconsider g1 = R_Cut ((Lower_Seq (C,n)),p) as being_S-Seq FinSequence of (TOP-REAL 2) by A1, JORDAN3:35;
len g1 in dom g1 by FINSEQ_5:6;
then A6: g1 /. (len g1) = g1 . (len g1) by PARTFUN1:def_6
.= p by A1, JORDAN3:24 ;
reconsider g = Rev g1 as being_S-Seq FinSequence of (TOP-REAL 2) ;
<*p*> is_in_the_area_of Cage (C,n) by A1, JORDAN1E:18, SPRECT_3:46;
then g1 is_in_the_area_of Cage (C,n) by A1, JORDAN1E:18, SPRECT_3:52;
then A7: g is_in_the_area_of Cage (C,n) by SPRECT_3:51;
A8: g /. 1 = g1 /. (len g1) by FINSEQ_5:65;
A9: g /. (len g) = g /. (len g1) by FINSEQ_5:def_3
.= g1 /. 1 by FINSEQ_5:65 ;
(g1 /. 1) `1 = ((Lower_Seq (C,n)) /. 1) `1 by A1, SPRECT_3:22
.= (E-max (L~ (Cage (C,n)))) `1 by JORDAN1F:6
.= E-bound (L~ (Cage (C,n))) by EUCLID:52 ;
then A10: g is_a_h.c._for Cage (C,n) by A2, A7, A8, A9, A6, SPRECT_2:def_2;
A11: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
1 in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A12: (Upper_Seq (C,n)) . 1 = W-min (L~ (Cage (C,n))) by A11, PARTFUN1:def_6;
A13: L~ g = L~ g1 by SPPOL_2:22;
len (Cage (C,n)) > 4 by GOBOARD7:34;
then A14: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2;
now__::_thesis:_contradiction
percases ( W-min (L~ (Cage (C,n))) <> SW-corner (L~ (Cage (C,n))) or W-min (L~ (Cage (C,n))) = SW-corner (L~ (Cage (C,n))) ) ;
supposeA15: W-min (L~ (Cage (C,n))) <> SW-corner (L~ (Cage (C,n))) ; ::_thesis: contradiction
A16: not SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n))
proof
A17: (SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
A18: (SW-corner (L~ (Cage (C,n)))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
then (SW-corner (L~ (Cage (C,n)))) `2 <= N-bound (L~ (Cage (C,n))) by SPRECT_1:22;
then SW-corner (L~ (Cage (C,n))) in { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = W-bound (L~ (Cage (C,n))) & p1 `2 <= N-bound (L~ (Cage (C,n))) & p1 `2 >= S-bound (L~ (Cage (C,n))) ) } by A17, A18;
then A19: SW-corner (L~ (Cage (C,n))) in LSeg ((SW-corner (L~ (Cage (C,n)))),(NW-corner (L~ (Cage (C,n))))) by SPRECT_1:26;
assume SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) ; ::_thesis: contradiction
then SW-corner (L~ (Cage (C,n))) in (LSeg ((SW-corner (L~ (Cage (C,n)))),(NW-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A14, A19, XBOOLE_0:def_4;
then A20: (SW-corner (L~ (Cage (C,n)))) `2 >= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;
A21: (W-min (L~ (Cage (C,n)))) `1 = (SW-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:29;
(W-min (L~ (Cage (C,n)))) `2 >= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;
then (W-min (L~ (Cage (C,n)))) `2 = (SW-corner (L~ (Cage (C,n)))) `2 by A20, XXREAL_0:1;
hence contradiction by A15, A21, TOPREAL3:6; ::_thesis: verum
end;
N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by Th7;
then R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) = mid ((Upper_Seq (C,n)),1,((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)))) by JORDAN1G:49;
then A22: rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119;
rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) by JORDAN1G:39;
then rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) c= rng (Cage (C,n)) by A22, XBOOLE_1:1;
then not SW-corner (L~ (Cage (C,n))) in rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A16;
then rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) misses {(SW-corner (L~ (Cage (C,n))))} by ZFMISC_1:50;
then rng (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) misses rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38;
then A23: rng (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) misses rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_5:57;
set h1 = (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>;
A24: <*(SW-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93;
W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13;
then A25: N-bound (L~ (Cage (C,n))) >= (W-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
A26: N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by Th7;
then <*(N-min (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by JORDAN1E:17, SPRECT_3:46;
then R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) is_in_the_area_of Cage (C,n) by A26, JORDAN1E:17, SPRECT_3:52;
then A27: Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51;
(W-max (L~ (Cage (C,n)))) `2 > (W-min (L~ (Cage (C,n)))) `2 by SPRECT_2:57;
then A28: N-min (L~ (Cage (C,n))) <> (Upper_Seq (C,n)) . 1 by A12, A25, EUCLID:52;
then reconsider RCutUS = R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) as being_S-Seq FinSequence of (TOP-REAL 2) by A26, JORDAN3:35;
A29: (Rev RCutUS) /. (len (Rev RCutUS)) = (Rev RCutUS) /. (len RCutUS) by FINSEQ_5:def_3
.= RCutUS /. 1 by FINSEQ_5:65
.= (Upper_Seq (C,n)) /. 1 by A26, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then ((Rev RCutUS) /. (len (Rev RCutUS))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= (SW-corner (L~ (Cage (C,n)))) `1 by EUCLID:52
.= (<*(SW-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_4:16 ;
then reconsider h1 = (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) ^ <*(SW-corner (L~ (Cage (C,n))))*> as one-to-one special FinSequence of (TOP-REAL 2) by A23, A24, FINSEQ_3:91, GOBOARD2:8;
set h = Rev h1;
A30: Rev h1 is special by SPPOL_2:40;
<*(SW-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:28;
then h1 is_in_the_area_of Cage (C,n) by A27, SPRECT_2:24;
then A31: Rev h1 is_in_the_area_of Cage (C,n) by SPRECT_3:51;
L~ (Rev h1) = L~ h1 by SPPOL_2:22;
then A32: L~ (Rev h1) = (L~ (Rev RCutUS)) \/ (LSeg (((Rev RCutUS) /. (len (Rev RCutUS))),(SW-corner (L~ (Cage (C,n)))))) by SPPOL_2:19;
A33: (Index ((N-min (L~ (Cage (C,n)))),(Upper_Seq (C,n)))) + 1 >= 0 + 1 by XREAL_1:7;
A34: 2 <= len g by TOPREAL1:def_8;
len h1 = (len (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))))) + 1 by FINSEQ_2:16
.= (len (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) + 1 by FINSEQ_5:def_3
.= ((Index ((N-min (L~ (Cage (C,n)))),(Upper_Seq (C,n)))) + 1) + 1 by A26, A28, JORDAN3:25 ;
then len h1 >= 1 + 1 by A33, XREAL_1:7;
then A35: len (Rev h1) >= 2 by FINSEQ_5:def_3;
A36: (Rev h1) /. 1 = h1 /. (len h1) by FINSEQ_5:65;
A37: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
1 in dom (Rev RCutUS) by FINSEQ_5:6;
then h1 /. 1 = (Rev RCutUS) /. 1 by FINSEQ_4:68
.= (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) /. (len (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) by FINSEQ_5:65
.= N-min (L~ (Cage (C,n))) by A26, Th45 ;
then A38: (h1 /. 1) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
A39: (h1 /. (len h1)) `2 = (h1 /. ((len (Rev (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))))) + 1)) `2 by FINSEQ_2:16
.= (SW-corner (L~ (Cage (C,n)))) `2 by FINSEQ_4:67
.= S-bound (L~ (Cage (C,n))) by EUCLID:52 ;
A40: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6;
(Rev h1) /. (len (Rev h1)) = (Rev h1) /. (len h1) by FINSEQ_5:def_3
.= h1 /. 1 by FINSEQ_5:65 ;
then Rev h1 is_a_v.c._for Cage (C,n) by A31, A38, A36, A39, SPRECT_2:def_3;
then L~ g meets L~ (Rev h1) by A10, A30, A34, A35, SPRECT_2:29;
then consider x being set such that
A41: x in L~ g and
A42: x in L~ (Rev h1) by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A41;
A43: L~ RCutUS c= L~ (Upper_Seq (C,n)) by Th7, JORDAN3:41;
A44: L~ g c= L~ (Lower_Seq (C,n)) by A1, A13, JORDAN3:41;
then A45: x in L~ (Lower_Seq (C,n)) by A41;
now__::_thesis:_contradiction
percases ( x in L~ (Rev RCutUS) or x in LSeg (((Rev RCutUS) /. (len (Rev RCutUS))),(SW-corner (L~ (Cage (C,n))))) ) by A42, A32, XBOOLE_0:def_3;
suppose x in L~ (Rev RCutUS) ; ::_thesis: contradiction
then A46: x in L~ RCutUS by SPPOL_2:22;
then x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A41, A44, A43, XBOOLE_0:def_4;
then A47: x in {(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} by JORDAN1E:16;
now__::_thesis:_contradiction
percases ( x = E-max (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) ) by A47, TARSKI:def_2;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A46, JORDAN1F:7;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A37, PARTFUN1:def_6;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by A26, Th43;
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by A37, PARTFUN1:def_6;
then A48: E-max (L~ (Cage (C,n))) = N-min (L~ (Cage (C,n))) by JORDAN1F:7;
N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:11;
then A49: E-bound (L~ (Cage (C,n))) >= (N-max (L~ (Cage (C,n)))) `1 by PSCOMP_1:24;
(N-max (L~ (Cage (C,n)))) `1 > (N-min (L~ (Cage (C,n)))) `1 by SPRECT_2:51;
hence contradiction by A48, A49, EUCLID:52; ::_thesis: verum
end;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A13, A41, JORDAN1F:8;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A40, PARTFUN1:def_6;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = p by A1, Th43;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = p by A40, PARTFUN1:def_6;
hence contradiction by A3, JORDAN1F:8; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA50: x in LSeg (((Rev RCutUS) /. (len (Rev RCutUS))),(SW-corner (L~ (Cage (C,n))))) ; ::_thesis: contradiction
(W-min (L~ (Cage (C,n)))) `2 >= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;
then A51: (W-min (L~ (Cage (C,n)))) `2 >= x `2 by A29, A50, TOPREAL1:4;
A52: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
(SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
then A53: x `1 = W-bound (L~ (Cage (C,n))) by A29, A50, A52, GOBOARD7:5;
L~ (Cage (C,n)) = (L~ (Lower_Seq (C,n))) \/ (L~ (Upper_Seq (C,n))) by JORDAN1E:13;
then L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
then x in W-most (L~ (Cage (C,n))) by A45, A53, SPRECT_2:12;
then x `2 >= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;
then x `2 = (W-min (L~ (Cage (C,n)))) `2 by A51, XXREAL_0:1;
then x = W-min (L~ (Cage (C,n))) by A52, A53, TOPREAL3:6;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A13, A41, JORDAN1F:8;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A40, PARTFUN1:def_6;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = p by A1, Th43;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = p by A40, PARTFUN1:def_6;
hence contradiction by A3, JORDAN1F:8; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA54: W-min (L~ (Cage (C,n))) = SW-corner (L~ (Cage (C,n))) ; ::_thesis: contradiction
set h = R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))));
A55: 2 <= len g by TOPREAL1:def_8;
W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13;
then A56: N-bound (L~ (Cage (C,n))) >= (W-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
A57: N-min (L~ (Cage (C,n))) in L~ (Upper_Seq (C,n)) by Th7;
then <*(N-min (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by JORDAN1E:17, SPRECT_3:46;
then A58: R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) is_in_the_area_of Cage (C,n) by A57, JORDAN1E:17, SPRECT_3:52;
(W-max (L~ (Cage (C,n)))) `2 > (W-min (L~ (Cage (C,n)))) `2 by SPRECT_2:57;
then N-min (L~ (Cage (C,n))) <> (Upper_Seq (C,n)) . 1 by A12, A56, EUCLID:52;
then reconsider RCutUS = R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) as being_S-Seq FinSequence of (TOP-REAL 2) by A57, JORDAN3:35;
A59: len RCutUS >= 2 by TOPREAL1:def_8;
(R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) /. (len (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))))) = N-min (L~ (Cage (C,n))) by A57, Th45;
then A60: ((R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) /. (len (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
RCutUS /. 1 = (Upper_Seq (C,n)) /. 1 by A57, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then ((R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) /. 1) `2 = S-bound (L~ (Cage (C,n))) by A54, EUCLID:52;
then R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n))))) is_a_v.c._for Cage (C,n) by A58, A60, SPRECT_2:def_3;
then L~ g meets L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A10, A55, A59, SPRECT_2:29;
then consider x being set such that
A61: x in L~ g and
A62: x in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A61;
A63: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6;
A64: L~ g c= L~ (Lower_Seq (C,n)) by A1, A13, JORDAN3:41;
A65: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
L~ RCutUS c= L~ (Upper_Seq (C,n)) by Th7, JORDAN3:41;
then x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A61, A62, A64, XBOOLE_0:def_4;
then A66: x in {(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} by JORDAN1E:16;
now__::_thesis:_contradiction
percases ( x = E-max (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) ) by A66, TARSKI:def_2;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A62, JORDAN1F:7;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) in L~ (R_Cut ((Upper_Seq (C,n)),(N-min (L~ (Cage (C,n)))))) by A65, PARTFUN1:def_6;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by A57, Th43;
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by A65, PARTFUN1:def_6;
then A67: E-max (L~ (Cage (C,n))) = N-min (L~ (Cage (C,n))) by JORDAN1F:7;
N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:11;
then A68: E-bound (L~ (Cage (C,n))) >= (N-max (L~ (Cage (C,n)))) `1 by PSCOMP_1:24;
(N-max (L~ (Cage (C,n)))) `1 > (N-min (L~ (Cage (C,n)))) `1 by SPRECT_2:51;
hence contradiction by A67, A68, EUCLID:52; ::_thesis: verum
end;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A13, A61, JORDAN1F:8;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) in L~ (R_Cut ((Lower_Seq (C,n)),p)) by A63, PARTFUN1:def_6;
then (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = p by A1, Th43;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = p by A63, PARTFUN1:def_6;
hence contradiction by A3, JORDAN1F:8; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
theorem :: JORDAN1J:48
for G being Go-board
for f, g being FinSequence of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < len f & f ^ g is_sequence_on G holds
( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) )
proof
let G be Go-board; ::_thesis: for f, g being FinSequence of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < len f & f ^ g is_sequence_on G holds
( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) )
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k < len f & f ^ g is_sequence_on G holds
( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) )
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < len f & f ^ g is_sequence_on G implies ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) ) )
assume that
A1: 1 <= k and
A2: k < len f and
A3: f ^ g is_sequence_on G ; ::_thesis: ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) )
A4: k + 1 <= len f by A2, NAT_1:13;
A5: (f ^ g) | (len f) = f by FINSEQ_5:23;
len f <= (len f) + (len g) by NAT_1:11;
then len f <= len (f ^ g) by FINSEQ_1:22;
then k + 1 <= len (f ^ g) by A4, XXREAL_0:2;
hence ( left_cell ((f ^ g),k,G) = left_cell (f,k,G) & right_cell ((f ^ g),k,G) = right_cell (f,k,G) ) by A1, A3, A5, A4, GOBRD13:31; ::_thesis: verum
end;
theorem Th49: :: JORDAN1J:49
for D being set
for f, g being FinSequence of D
for i being Element of NAT st i <= len f holds
(f ^' g) | i = f | i
proof
let D be set ; ::_thesis: for f, g being FinSequence of D
for i being Element of NAT st i <= len f holds
(f ^' g) | i = f | i
let f, g be FinSequence of D; ::_thesis: for i being Element of NAT st i <= len f holds
(f ^' g) | i = f | i
let i be Element of NAT ; ::_thesis: ( i <= len f implies (f ^' g) | i = f | i )
assume A1: i <= len f ; ::_thesis: (f ^' g) | i = f | i
then A2: len (f | i) = i by FINSEQ_1:59;
percases ( g <> {} or g = {} ) ;
supposeA3: g <> {} ; ::_thesis: (f ^' g) | i = f | i
then len g >= 0 + 1 by NAT_1:13;
then i + 1 <= (len f) + (len g) by A1, XREAL_1:7;
then i + 1 <= (len (f ^' g)) + 1 by A3, GRAPH_2:13;
then i <= len (f ^' g) by XREAL_1:6;
then A4: len ((f ^' g) | i) = i by FINSEQ_1:59;
then A5: dom ((f ^' g) | i) = Seg i by FINSEQ_1:def_3;
now__::_thesis:_for_j_being_Nat_st_j_in_dom_((f_^'_g)_|_i)_holds_
((f_^'_g)_|_i)_._j_=_(f_|_i)_._j
let j be Nat; ::_thesis: ( j in dom ((f ^' g) | i) implies ((f ^' g) | i) . j = (f | i) . j )
A6: j in NAT by ORDINAL1:def_12;
assume A7: j in dom ((f ^' g) | i) ; ::_thesis: ((f ^' g) | i) . j = (f | i) . j
then A8: 1 <= j by A5, FINSEQ_1:1;
j <= i by A5, A7, FINSEQ_1:1;
then A9: j <= len f by A1, XXREAL_0:2;
thus ((f ^' g) | i) . j = ((f ^' g) | (Seg i)) . j by FINSEQ_1:def_15
.= (f ^' g) . j by A5, A7, FUNCT_1:49
.= f . j by A6, A8, A9, GRAPH_2:14
.= (f | (Seg i)) . j by A5, A7, FUNCT_1:49
.= (f | i) . j by FINSEQ_1:def_15 ; ::_thesis: verum
end;
hence (f ^' g) | i = f | i by A2, A4, FINSEQ_2:9; ::_thesis: verum
end;
suppose g = {} ; ::_thesis: (f ^' g) | i = f | i
hence (f ^' g) | i = f | i by GRAPH_2:55; ::_thesis: verum
end;
end;
end;
theorem Th50: :: JORDAN1J:50
for D being set
for f, g being FinSequence of D holds (f ^' g) | (len f) = f
proof
let D be set ; ::_thesis: for f, g being FinSequence of D holds (f ^' g) | (len f) = f
let f, g be FinSequence of D; ::_thesis: (f ^' g) | (len f) = f
f | (len f) = f | (Seg (len f)) by FINSEQ_1:def_15;
hence (f ^' g) | (len f) = f by Th49, FINSEQ_2:20; ::_thesis: verum
end;
theorem Th51: :: JORDAN1J:51
for G being Go-board
for f, g being FinSequence of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < len f & f ^' g is_sequence_on G holds
( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) )
proof
let G be Go-board; ::_thesis: for f, g being FinSequence of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < len f & f ^' g is_sequence_on G holds
( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) )
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k < len f & f ^' g is_sequence_on G holds
( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) )
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < len f & f ^' g is_sequence_on G implies ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) ) )
assume that
A1: 1 <= k and
A2: k < len f and
A3: f ^' g is_sequence_on G ; ::_thesis: ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) )
A4: k + 1 <= len f by A2, NAT_1:13;
A5: (f ^' g) | (len f) = f by Th50;
len f <= len (f ^' g) by TOPREAL8:7;
then k + 1 <= len (f ^' g) by A4, XXREAL_0:2;
hence ( left_cell ((f ^' g),k,G) = left_cell (f,k,G) & right_cell ((f ^' g),k,G) = right_cell (f,k,G) ) by A1, A3, A5, A4, GOBRD13:31; ::_thesis: verum
end;
theorem Th52: :: JORDAN1J:52
for G being Go-board
for f being S-Sequence_in_R2
for p being Point of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G & p in rng f holds
( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) )
proof
let G be Go-board; ::_thesis: for f being S-Sequence_in_R2
for p being Point of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G & p in rng f holds
( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) )
let f be S-Sequence_in_R2; ::_thesis: for p being Point of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G & p in rng f holds
( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) )
let p be Point of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G & p in rng f holds
( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) )
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < p .. f & f is_sequence_on G & p in rng f implies ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) )
assume that
A1: 1 <= k and
A2: k < p .. f and
A3: f is_sequence_on G and
A4: p in rng f ; ::_thesis: ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) )
A5: f | (p .. f) = mid (f,1,(p .. f)) by A1, A2, FINSEQ_6:116, XXREAL_0:2
.= R_Cut (f,p) by A4, JORDAN1G:49 ;
A6: k + 1 <= p .. f by A2, NAT_1:13;
p .. f <= len f by A4, FINSEQ_4:21;
then k + 1 <= len f by A6, XXREAL_0:2;
hence ( left_cell ((R_Cut (f,p)),k,G) = left_cell (f,k,G) & right_cell ((R_Cut (f,p)),k,G) = right_cell (f,k,G) ) by A1, A3, A5, A6, GOBRD13:31; ::_thesis: verum
end;
theorem Th53: :: JORDAN1J:53
for G being Go-board
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G holds
( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) )
proof
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G holds
( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G holds
( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) )
let p be Point of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k < p .. f & f is_sequence_on G holds
( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) )
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < p .. f & f is_sequence_on G implies ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) )
assume that
A1: 1 <= k and
A2: k < p .. f and
A3: f is_sequence_on G ; ::_thesis: ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) )
A4: k + 1 <= p .. f by A2, NAT_1:13;
A5: f | (p .. f) = f -: p by FINSEQ_5:def_1;
percases ( p in rng f or p .. f = 0 ) by TOPREAL8:4;
suppose p in rng f ; ::_thesis: ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) )
then p .. f <= len f by FINSEQ_4:21;
then k + 1 <= len f by A4, XXREAL_0:2;
hence ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) by A1, A3, A5, A4, GOBRD13:31; ::_thesis: verum
end;
suppose p .. f = 0 ; ::_thesis: ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) )
hence ( left_cell ((f -: p),k,G) = left_cell (f,k,G) & right_cell ((f -: p),k,G) = right_cell (f,k,G) ) by A2; ::_thesis: verum
end;
end;
end;
theorem Th54: :: JORDAN1J:54
for f, g being FinSequence of (TOP-REAL 2) st f is unfolded & f is s.n.c. & f is one-to-one & g is unfolded & g is s.n.c. & g is one-to-one & f /. (len f) = g /. 1 & (L~ f) /\ (L~ g) = {(g /. 1)} holds
f ^' g is s.n.c.
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f is unfolded & f is s.n.c. & f is one-to-one & g is unfolded & g is s.n.c. & g is one-to-one & f /. (len f) = g /. 1 & (L~ f) /\ (L~ g) = {(g /. 1)} implies f ^' g is s.n.c. )
assume that
A1: ( f is unfolded & f is s.n.c. & f is one-to-one ) and
A2: ( g is unfolded & g is s.n.c. & g is one-to-one ) and
A3: f /. (len f) = g /. 1 and
A4: (L~ f) /\ (L~ g) = {(g /. 1)} ; ::_thesis: f ^' g is s.n.c.
now__::_thesis:_for_i,_j_being_Nat_st_i_+_1_<_j_holds_
LSeg_((f_^'_g),i)_misses_LSeg_((f_^'_g),j)
let i, j be Nat; ::_thesis: ( i + 1 < j implies LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) )
assume A5: i + 1 < j ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j)
A6: i in NAT by ORDINAL1:def_12;
A7: j in NAT by ORDINAL1:def_12;
now__::_thesis:_LSeg_((f_^'_g),i)_misses_LSeg_((f_^'_g),j)
percases ( j < len f or j >= len f ) ;
supposeA8: j < len f ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j)
then i + 1 < len f by A5, XXREAL_0:2;
then i < len f by NAT_1:13;
then A9: LSeg ((f ^' g),i) = LSeg (f,i) by A6, TOPREAL8:28;
LSeg ((f ^' g),j) = LSeg (f,j) by A7, A8, TOPREAL8:28;
hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) by A1, A5, A9, TOPREAL1:def_7; ::_thesis: verum
end;
suppose j >= len f ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j)
then consider k being Nat such that
A10: j = (len f) + k by NAT_1:10;
A11: now__::_thesis:_not_g_is_trivial
assume g is trivial ; ::_thesis: contradiction
then len g < 2 by NAT_D:60;
then ( len g = 0 or len g = 1 ) by NAT_1:23;
then L~ g = {} by TOPREAL1:22;
hence contradiction by A4; ::_thesis: verum
end;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
A12: now__::_thesis:_not_f_is_empty
assume f is empty ; ::_thesis: contradiction
then len f = 0 ;
then L~ f = {} by TOPREAL1:22;
hence contradiction by A4; ::_thesis: verum
end;
now__::_thesis:_LSeg_((f_^'_g),i)_misses_LSeg_((f_^'_g),j)
percases ( ( i >= 1 & j + 1 <= len (f ^' g) ) or j + 1 > len (f ^' g) or i < 1 ) ;
supposeA13: ( i >= 1 & j + 1 <= len (f ^' g) ) ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j)
then j + 1 < (len (f ^' g)) + 1 by NAT_1:13;
then (len f) + (k + 1) < (len f) + (len g) by A10, A11, GRAPH_2:13;
then A14: k + 1 < len g by XREAL_1:7;
then A15: LSeg ((f ^' g),((len f) + k)) = LSeg (g,(k + 1)) by A3, A12, A11, TOPREAL8:31;
then A16: LSeg ((f ^' g),j) c= L~ g by A10, TOPREAL3:19;
now__::_thesis:_not_LSeg_((f_^'_g),i)_meets_LSeg_((f_^'_g),j)
percases ( i < len f or i >= len f ) ;
supposeA17: i < len f ; ::_thesis: not LSeg ((f ^' g),i) meets LSeg ((f ^' g),j)
then A18: i + 1 <= len f by NAT_1:13;
i + 1 > 1 by A13, NAT_1:13;
then A19: i + 1 in dom f by A18, FINSEQ_3:25;
A20: len g >= 2 by A11, NAT_D:60;
A21: LSeg ((f ^' g),i) = LSeg (f,i) by A6, A17, TOPREAL8:28;
then LSeg ((f ^' g),i) c= L~ f by TOPREAL3:19;
then A22: (LSeg ((f ^' g),i)) /\ (LSeg ((f ^' g),j)) c= {(g /. 1)} by A4, A16, XBOOLE_1:27;
assume LSeg ((f ^' g),i) meets LSeg ((f ^' g),j) ; ::_thesis: contradiction
then consider x being set such that
A23: x in LSeg ((f ^' g),i) and
A24: x in LSeg ((f ^' g),j) by XBOOLE_0:3;
x in (LSeg ((f ^' g),i)) /\ (LSeg ((f ^' g),j)) by A23, A24, XBOOLE_0:def_4;
then A25: x = g /. 1 by A22, TARSKI:def_1;
i in dom f by A13, A17, FINSEQ_3:25;
then (len f) + 0 < (len f) + k by A1, A3, A5, A10, A21, A23, A25, A19, GOBOARD2:2;
then k > 0 ;
then k + 1 > 0 + 1 by XREAL_1:6;
hence contradiction by A2, A10, A15, A24, A25, A20, JORDAN5B:30; ::_thesis: verum
end;
suppose i >= len f ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j)
then consider l being Nat such that
A26: i = (len f) + l by NAT_1:10;
reconsider l = l as Element of NAT by ORDINAL1:def_12;
(len f) + (l + 1) < (len f) + k by A5, A10, A26;
then l + 1 < k by XREAL_1:7;
then A27: (l + 1) + 1 < k + 1 by XREAL_1:6;
then (l + 1) + 1 < len g by A14, XXREAL_0:2;
then l + 1 < len g by NAT_1:13;
then LSeg ((f ^' g),((len f) + l)) = LSeg (g,(l + 1)) by A3, A12, A11, TOPREAL8:31;
hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) by A2, A10, A15, A26, A27, TOPREAL1:def_7; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) ; ::_thesis: verum
end;
suppose j + 1 > len (f ^' g) ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j)
then LSeg ((f ^' g),j) = {} by TOPREAL1:def_3;
hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) by XBOOLE_1:65; ::_thesis: verum
end;
suppose i < 1 ; ::_thesis: LSeg ((f ^' g),i) misses LSeg ((f ^' g),j)
then LSeg ((f ^' g),i) = {} by TOPREAL1:def_3;
hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) by XBOOLE_1:65; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) ; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^' g),i) misses LSeg ((f ^' g),j) ; ::_thesis: verum
end;
hence f ^' g is s.n.c. by TOPREAL1:def_7; ::_thesis: verum
end;
theorem Th55: :: JORDAN1J:55
for f, g being FinSequence of (TOP-REAL 2) st f is one-to-one & g is one-to-one & (rng f) /\ (rng g) c= {(g /. 1)} holds
f ^' g is one-to-one
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f is one-to-one & g is one-to-one & (rng f) /\ (rng g) c= {(g /. 1)} implies f ^' g is one-to-one )
assume that
A1: f is one-to-one and
A2: g is one-to-one and
A3: (rng f) /\ (rng g) c= {(g /. 1)} ; ::_thesis: f ^' g is one-to-one
percases ( rng g <> {} or rng g = {} ) ;
supposeA4: rng g <> {} ; ::_thesis: f ^' g is one-to-one
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_i_in_dom_(f_^'_g)_&_j_in_dom_(f_^'_g)_&_(f_^'_g)_/._i_=_(f_^'_g)_/._j_holds_
i_=_j
A5: (len (f ^' g)) + 1 = (len f) + (len g) by A4, GRAPH_2:13, RELAT_1:38;
let i, j be Element of NAT ; ::_thesis: ( i in dom (f ^' g) & j in dom (f ^' g) & (f ^' g) /. i = (f ^' g) /. j implies i = j )
assume that
A6: i in dom (f ^' g) and
A7: j in dom (f ^' g) and
A8: (f ^' g) /. i = (f ^' g) /. j ; ::_thesis: i = j
A9: 1 <= i by A6, FINSEQ_3:25;
j <= len (f ^' g) by A7, FINSEQ_3:25;
then A10: j < (len f) + (len g) by A5, NAT_1:13;
A11: len f = (len f) + 0 ;
i <= len (f ^' g) by A6, FINSEQ_3:25;
then A12: i < (len f) + (len g) by A5, NAT_1:13;
A13: 1 <= j by A7, FINSEQ_3:25;
A14: 1 in dom g by A4, FINSEQ_3:32;
now__::_thesis:_i_=_j
percases ( ( i <= len f & j <= len f ) or ( i > len f & j > len f ) or ( i <= len f & j > len f ) or ( j <= len f & i > len f ) ) ;
supposeA15: ( i <= len f & j <= len f ) ; ::_thesis: i = j
then A16: i in dom f by A9, FINSEQ_3:25;
A17: (f ^' g) /. j = f /. j by A13, A15, GRAPH_2:57;
A18: j in dom f by A13, A15, FINSEQ_3:25;
(f ^' g) /. i = f /. i by A9, A15, GRAPH_2:57;
hence i = j by A1, A8, A17, A16, A18, PARTFUN2:10; ::_thesis: verum
end;
supposeA19: ( i > len f & j > len f ) ; ::_thesis: i = j
then consider l being Nat such that
A20: j = (len f) + l by NAT_1:10;
consider k being Nat such that
A21: i = (len f) + k by A19, NAT_1:10;
reconsider k = k, l = l as Element of NAT by ORDINAL1:def_12;
l > 0 by A11, A19, A20;
then A22: l >= 0 + 1 by NAT_1:13;
then A23: l + 1 > 1 by NAT_1:13;
k > 0 by A11, A19, A21;
then A24: k >= 0 + 1 by NAT_1:13;
then A25: k + 1 > 1 by NAT_1:13;
A26: l < len g by A10, A20, XREAL_1:7;
then A27: (f ^' g) /. j = g /. (l + 1) by A20, A22, GRAPH_2:58;
A28: k < len g by A12, A21, XREAL_1:7;
then k + 1 <= len g by NAT_1:13;
then A29: k + 1 in dom g by A25, FINSEQ_3:25;
l + 1 <= len g by A26, NAT_1:13;
then A30: l + 1 in dom g by A23, FINSEQ_3:25;
(f ^' g) /. i = g /. (k + 1) by A21, A28, A24, GRAPH_2:58;
then k + 1 = l + 1 by A2, A8, A27, A29, A30, PARTFUN2:10;
hence i = j by A21, A20; ::_thesis: verum
end;
supposeA31: ( i <= len f & j > len f ) ; ::_thesis: i = j
then A32: i in dom f by A9, FINSEQ_3:25;
(f ^' g) /. i = f /. i by A9, A31, GRAPH_2:57;
then A33: (f ^' g) /. i in rng f by A32, PARTFUN2:2;
consider l being Nat such that
A34: j = (len f) + l by A31, NAT_1:10;
reconsider l = l as Element of NAT by ORDINAL1:def_12;
l > 0 by A11, A31, A34;
then A35: l >= 0 + 1 by NAT_1:13;
then A36: l + 1 > 1 by NAT_1:13;
A37: l < len g by A10, A34, XREAL_1:7;
then l + 1 <= len g by NAT_1:13;
then A38: l + 1 in dom g by A36, FINSEQ_3:25;
A39: (f ^' g) /. j = g /. (l + 1) by A34, A37, A35, GRAPH_2:58;
then (f ^' g) /. j in rng g by A38, PARTFUN2:2;
then (f ^' g) /. j in (rng f) /\ (rng g) by A8, A33, XBOOLE_0:def_4;
then g /. (l + 1) = g /. 1 by A3, A39, TARSKI:def_1;
hence i = j by A2, A14, A36, A38, PARTFUN2:10; ::_thesis: verum
end;
supposeA40: ( j <= len f & i > len f ) ; ::_thesis: i = j
then A41: j in dom f by A13, FINSEQ_3:25;
(f ^' g) /. j = f /. j by A13, A40, GRAPH_2:57;
then A42: (f ^' g) /. j in rng f by A41, PARTFUN2:2;
consider l being Nat such that
A43: i = (len f) + l by A40, NAT_1:10;
reconsider l = l as Element of NAT by ORDINAL1:def_12;
l > 0 by A11, A40, A43;
then A44: l >= 0 + 1 by NAT_1:13;
then A45: l + 1 > 1 by NAT_1:13;
A46: l < len g by A12, A43, XREAL_1:7;
then l + 1 <= len g by NAT_1:13;
then A47: l + 1 in dom g by A45, FINSEQ_3:25;
A48: (f ^' g) /. i = g /. (l + 1) by A43, A46, A44, GRAPH_2:58;
then (f ^' g) /. i in rng g by A47, PARTFUN2:2;
then (f ^' g) /. i in (rng f) /\ (rng g) by A8, A42, XBOOLE_0:def_4;
then g /. (l + 1) = g /. 1 by A3, A48, TARSKI:def_1;
hence i = j by A2, A14, A45, A47, PARTFUN2:10; ::_thesis: verum
end;
end;
end;
hence i = j ; ::_thesis: verum
end;
hence f ^' g is one-to-one by PARTFUN2:9; ::_thesis: verum
end;
suppose rng g = {} ; ::_thesis: f ^' g is one-to-one
then g = {} by RELAT_1:41;
hence f ^' g is one-to-one by A1, GRAPH_2:55; ::_thesis: verum
end;
end;
end;
theorem Th56: :: JORDAN1J:56
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in rng f & p <> f . 1 holds
(Index (p,f)) + 1 = p .. f
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in rng f & p <> f . 1 holds
(Index (p,f)) + 1 = p .. f
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in rng f & p <> f . 1 implies (Index (p,f)) + 1 = p .. f )
assume that
A1: f is being_S-Seq and
A2: p in rng f and
A3: p <> f . 1 ; ::_thesis: (Index (p,f)) + 1 = p .. f
A4: 1 <= p .. f by A2, FINSEQ_4:21;
p .. f <> 1 by A2, A3, FINSEQ_4:19;
then A5: 1 < p .. f by A4, XXREAL_0:1;
A6: f . (p .. f) = p by A2, FINSEQ_4:19;
p .. f <= len f by A2, FINSEQ_4:21;
hence (Index (p,f)) + 1 = p .. f by A1, A5, A6, JORDAN3:12; ::_thesis: verum
end;
theorem Th57: :: JORDAN1J:57
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
j <> k
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
j <> k
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) holds
j <> k
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) implies j <> k )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: k <= width (Gauge (C,n)) and
A5: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) and
A6: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) and
A7: j = k ; ::_thesis: contradiction
A8: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A4, A7, MATRIX_1:36;
(Gauge (C,n)) * (i,k) in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A5, A6, A7, XBOOLE_0:def_4;
then A9: (Gauge (C,n)) * (i,k) in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
A10: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A11: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A12: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A3, A4, A7, MATRIX_1:36;
A13: [1,j] in Indices (Gauge (C,n)) by A3, A4, A7, A11, MATRIX_1:36;
percases ( (Gauge (C,n)) * (i,k) = W-min (L~ (Cage (C,n))) or (Gauge (C,n)) * (i,k) = E-max (L~ (Cage (C,n))) ) by A9, TARSKI:def_2;
supposeA14: (Gauge (C,n)) * (i,k) = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
((Gauge (C,n)) * (1,j)) `1 = W-bound (L~ (Cage (C,n))) by A3, A4, A7, A10, JORDAN1A:73;
then (W-min (L~ (Cage (C,n)))) `1 <> W-bound (L~ (Cage (C,n))) by A1, A7, A8, A13, A14, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
supposeA15: (Gauge (C,n)) * (i,k) = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A3, A4, A7, A10, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A7, A8, A12, A15, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
end;
end;
theorem Th58: :: JORDAN1J:58
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C )
set Ga = Gauge (C,n);
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
set LA = Lower_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gij = (Gauge (C,n)) * (i,j);
set Gik = (Gauge (C,n)) * (i,k);
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} and
A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} and
A8: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) misses Lower_Arc C ; ::_thesis: contradiction
(Gauge (C,n)) * (i,j) in {((Gauge (C,n)) * (i,j))} by TARSKI:def_1;
then A9: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:def_4;
(Gauge (C,n)) * (i,k) in {((Gauge (C,n)) * (i,k))} by TARSKI:def_1;
then A10: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:def_4;
then A11: j <> k by A1, A2, A3, A5, A9, Th57;
A12: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
A13: 1 <= k by A3, A4, XXREAL_0:2;
A14: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A12, MATRIX_1:36;
A15: [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, A13, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)));
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)));
A16: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A17: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then 1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A18: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A19: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A5, A13, A16, JORDAN1A:73 ;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A20: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A21: [1,k] in Indices (Gauge (C,n)) by A5, A13, MATRIX_1:36;
then A22: (Gauge (C,n)) * (i,k) <> (Upper_Seq (C,n)) . 1 by A1, A15, A18, A19, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
A23: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A24: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A25: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A24, FINSEQ_3:25;
then A26: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
A27: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A5, A13, A16, JORDAN1A:73 ;
A28: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A12, MATRIX_1:36;
then A29: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A21, A26, A27, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34;
A30: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A5, A13, A20, MATRIX_1:36;
A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A5, A13, A16, JORDAN1A:71 ;
then A32: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . 1 by A2, A28, A30, A31, JORDAN1G:7;
A33: len go >= 1 + 1 by TOPREAL1:def_8;
A34: (Gauge (C,n)) * (i,k) in rng (Upper_Seq (C,n)) by A1, A2, A5, A10, A13, Th40, JORDAN1G:4;
then A35: go is_sequence_on Gauge (C,n) by Th38, JORDAN1G:4;
A36: len do >= 1 + 1 by TOPREAL1:def_8;
A37: (Gauge (C,n)) * (i,j) in rng (Lower_Seq (C,n)) by A1, A2, A3, A9, A12, Th40, JORDAN1G:5;
then A38: do is_sequence_on Gauge (C,n) by Th39, JORDAN1G:5;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5;
A39: len go > 1 by A33, NAT_1:13;
then A40: len go in dom go by FINSEQ_3:25;
then A41: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,k) by A10, JORDAN3:24 ;
len do >= 1 by A36, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A42: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,j) by A9, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A40, FINSEQ_3:26;
A43: m + 1 = len go ;
then A44: (len go) -' 1 = m by NAT_D:34;
A45: LSeg (go,m) c= L~ go by TOPREAL3:19;
A46: L~ go c= L~ (Upper_Seq (C,n)) by A10, JORDAN3:41;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A45, XBOOLE_1:1;
then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,k))} by A6, XBOOLE_1:26;
m >= 1 by A33, XREAL_1:19;
then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i,k))) by A41, A43, TOPREAL1:def_3;
{((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
A49: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
assume x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A50: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1;
(Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A50, A49, XBOOLE_0:def_4; ::_thesis: verum
end;
then A51: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,k))} by A47, XBOOLE_0:def_10;
A52: LSeg (do,1) c= L~ do by TOPREAL3:19;
A53: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A52, XBOOLE_1:1;
then A54: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,j))} by A7, XBOOLE_1:26;
A55: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i,j)),(do /. (1 + 1))) by A36, A42, TOPREAL1:def_3;
{((Gauge (C,n)) * (i,j))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
A56: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
assume x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A57: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1;
(Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A57, A56, XBOOLE_0:def_4; ::_thesis: verum
end;
then A58: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A54, XBOOLE_0:def_10;
A59: go /. 1 = (Upper_Seq (C,n)) /. 1 by A10, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A60: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A9, Th35 ;
A61: rng go c= L~ go by A33, SPPOL_2:18;
A62: rng do c= L~ do by A36, SPPOL_2:18;
A63: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A64: x = go /. 1 by TARSKI:def_1;
then A65: x in rng go by FINSEQ_6:42;
x in rng do by A60, A64, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A3, A12, A20, MATRIX_1:36;
(L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A68: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A69: x in L~ do by XBOOLE_0:def_4;
A70: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A71: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,j) by A9, A66, A69, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A3, A12, A16, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A14, A67, A71, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
x in L~ go by A68, XBOOLE_0:def_4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A46, A53, A69, XBOOLE_0:def_4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
hence x in {(go /. 1)} by A59, A70, TARSKI:def_1; ::_thesis: verum
end;
then A72: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def_10;
set W2 = go /. 2;
A73: 2 in dom go by A33, FINSEQ_3:25;
A74: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume ((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A5, A13, A16, JORDAN1A:73;
hence contradiction by A1, A15, A21, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)))) by A34, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n))) by A34, FINSEQ_4:21, FINSEQ_6:116 ;
then A75: go /. 2 = (Upper_Seq (C,n)) /. 2 by A73, FINSEQ_4:70;
set pion = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>;
A76: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) )
then n in Seg 2 by FINSEQ_1:89;
then ( n = 1 or n = 2 ) by FINSEQ_1:2, TARSKI:def_2;
hence ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A14, A15, FINSEQ_4:17; ::_thesis: verum
end;
A77: (Gauge (C,n)) * (i,k) <> (Gauge (C,n)) * (i,j) by A11, A14, A15, GOBOARD1:5;
A78: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A5, A13, GOBOARD5:2
.= ((Gauge (C,n)) * (i,j)) `1 by A1, A2, A3, A12, GOBOARD5:2 ;
then LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) is vertical by SPPOL_1:16;
then <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> is being_S-Seq by A77, JORDAN1B:7;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A79: pion1 is_sequence_on Gauge (C,n) and
A80: pion1 is being_S-Seq and
A81: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = L~ pion1 and
A82: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 1 = pion1 /. 1 and
A83: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = pion1 /. (len pion1) and
A84: len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> <= len pion1 by A76, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A80;
set godo = (go ^' pion1) ^' do;
A85: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A86: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A87: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2;
then A88: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A89: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A90: 1 + 1 <= len ((go ^' pion1) ^' do) by A87, XXREAL_0:2;
A91: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A92: go /. (len go) = pion1 /. 1 by A41, A82, FINSEQ_4:17;
then A93: go ^' pion1 is_sequence_on Gauge (C,n) by A35, A79, TOPREAL8:12;
A94: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) by A83, GRAPH_2:54
.= <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
then A95: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A38, A93, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A81, TOPREAL3:19;
then LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A96: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i,k))} by A44, A51, XBOOLE_1:27;
A97: len pion1 >= 1 + 1 by A84, FINSEQ_1:44;
{((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A98: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1;
A99: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
(Gauge (C,n)) * (i,k) in LSeg (pion1,1) by A41, A92, A97, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A98, A99, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A96, XBOOLE_0:def_10;
then A100: go ^' pion1 is unfolded by A92, TOPREAL8:34;
len pion1 >= 2 + 0 by A84, FINSEQ_1:44;
then A101: (len pion1) - 2 >= 0 by XREAL_1:19;
((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A101, XREAL_0:def_2 ;
then A102: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A103: (len pion1) - 1 >= 1 by A97, XREAL_1:19;
then A104: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A105: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A101, XREAL_0:def_2
.= (len pion1) -' 1 by A103, XREAL_0:def_2 ;
((len pion1) - 1) + 1 <= len pion1 ;
then A106: (len pion1) -' 1 < len pion1 by A104, NAT_1:13;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A81, TOPREAL3:19;
then LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A107: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i,j))} by A58, XBOOLE_1:27;
{((Gauge (C,n)) * (i,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A108: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by A83, A104, FINSEQ_1:44
.= (Gauge (C,n)) * (i,j) by FINSEQ_4:17 ;
then A109: (Gauge (C,n)) * (i,j) in LSeg (pion1,((len pion1) -' 1)) by A103, A104, TOPREAL1:21;
(Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A108, A109, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A107, XBOOLE_0:def_10;
then A110: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A92, A94, A105, A106, TOPREAL8:31;
A111: not go ^' pion1 is trivial by A87, NAT_D:60;
A112: rng pion1 c= L~ pion1 by A97, SPPOL_2:18;
A113: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A114: x = pion1 /. 1 by TARSKI:def_1;
then A115: x in rng pion1 by FINSEQ_6:42;
x in rng go by A92, A114, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A61, A112, A115, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A116: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A117: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ go by A116, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A117, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A6, A41, A81, A92, SPPOL_2:21; ::_thesis: verum
end;
then A118: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A113, XBOOLE_0:def_10;
then A119: go ^' pion1 is s.n.c. by A92, Th54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1:27;
then A120: go ^' pion1 is one-to-one by Th55;
A121: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
A122: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A123: x = pion1 /. (len pion1) by TARSKI:def_1;
then A124: x in rng pion1 by REVROT_1:3;
x in rng do by A83, A121, A123, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A62, A112, A124, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A125: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A126: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A125, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A126, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A7, A42, A81, A83, A121, SPPOL_2:21; ::_thesis: verum
end;
then A127: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A122, XBOOLE_0:def_10;
A128: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A92, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A72, A83, A121, A127, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
do /. (len do) = (go ^' pion1) /. 1 by A60, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A90, A94, A95, A100, A102, A110, A111, A119, A120, A128, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A129: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def_9;
then A130: Lower_Arc C is connected by JORDAN6:10;
A131: W-min C in Lower_Arc C by A129, TOPREAL1:1;
A132: E-max C in Lower_Arc C by A129, TOPREAL1:1;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A133: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A134: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, SPRECT_5:22;
then (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:23, XXREAL_0:2;
then (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:24, XXREAL_0:2;
then A135: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:25, XXREAL_0:2;
A136: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_.._(Upper_Seq_(C,n))_<=_1
assume A137: ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) >= 1 by A34, FINSEQ_4:21;
then ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) = 1 by A137, XXREAL_0:1;
then (Gauge (C,n)) * (i,k) = (Upper_Seq (C,n)) /. 1 by A34, FINSEQ_5:38;
hence contradiction by A18, A22, JORDAN1F:5; ::_thesis: verum
end;
A138: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A139: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A140: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A90, A95, JORDAN9:27;
A141: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A94, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A92, TOPREAL8:35 ;
A142: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A143: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A144: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A142, XBOOLE_1:7;
A145: L~ go c= L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1;
A146: L~ do c= L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1;
A147: W-min C in C by SPRECT_1:13;
A148: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
A149: now__::_thesis:_not_W-min_C_in_L~_godo
assume W-min C in L~ godo ; ::_thesis: contradiction
then A150: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A141, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A150, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A145, A147, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A81, A131, A148, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A146, A147, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A86, JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A135, A139, Th53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)))),1,(Gauge (C,n))) by A34, A91, A136, Th52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A93, Th51
.= right_cell (godo,1,(Gauge (C,n))) by A88, A95, Th51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A151: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A149, XBOOLE_0:def_5;
A152: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A59, GRAPH_2:53 ;
A153: len (Upper_Seq (C,n)) >= 2 by A17, XXREAL_0:2;
A154: godo /. 2 = (go ^' pion1) /. 2 by A87, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A33, A75, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A153, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A155: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42;
then W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A61, XBOOLE_0:def_3;
then A156: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A145, A146, A155, Th21, XBOOLE_1:8;
A157: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A158: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
W-bound (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = ((Gauge (C,n)) * (i,k)) `1 by A78, SPRECT_1:54;
then A159: W-bound (L~ pion1) = ((Gauge (C,n)) * (i,k)) `1 by A81, SPPOL_2:21;
((Gauge (C,n)) * (i,k)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A143, PSCOMP_1:24;
then ((Gauge (C,n)) * (i,k)) `1 > W-bound (L~ (Cage (C,n))) by A74, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A155, A156, A157, A158, A159, Th33;
then A160: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A141, A156, XBOOLE_1:4;
A161: rng godo c= L~ godo by A87, A89, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A90, FINSEQ_3:25;
then A162: godo /. 2 in rng godo by PARTFUN2:2;
godo /. 2 in W-most (L~ (Cage (C,n))) by A154, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A160, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A161, A162, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A160, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A163: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A164: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A165: p in east_halfline (E-max C) and
A166: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A165;
p in L~ (Upper_Seq (C,n)) by A46, A166;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A143, A165, XBOOLE_0:def_4;
then A167: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A168: p = E-max (L~ (Cage (C,n))) by A46, A166, Th46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,k) by A10, A163, A166, Th43;
then ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A5, A13, A16, A167, A168, JORDAN1A:71;
hence contradiction by A2, A15, A30, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A169: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A141, XBOOLE_1:70;
percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A169, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A164; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A170: p in east_halfline (E-max C) and
A171: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A170;
A172: p `2 = (E-max C) `2 by A170, TOPREAL1:def_11;
i + 1 <= len (Gauge (C,n)) by A2, NAT_1:13;
then (i + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A173: i <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
A174: (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
p `1 = ((Gauge (C,n)) * (i,k)) `1 by A78, A81, A148, A171, GOBOARD7:5;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A1, A5, A13, A16, A20, A173, A174, JORDAN1A:18;
then p `1 <= E-bound C by A20, JORDAN8:12;
then A175: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A170, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A175, XXREAL_0:1;
then p = E-max C by A172, TOPREAL3:6;
hence contradiction by A8, A81, A132, A148, A171, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A176: p in east_halfline (E-max C) and
A177: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A176;
A178: (E-max C) `2 = p `2 by A176, TOPREAL1:def_11;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A179: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider t being Nat such that
A180: t in dom (Lower_Seq (C,n)) and
A181: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i,j) by A37, FINSEQ_2:10;
1 <= t by A180, FINSEQ_3:25;
then A182: 1 < t by A32, A181, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A180, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = t by A181, A182, JORDAN3:12;
then A183: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A9, A181, JORDAN3:26;
Index (p,do) < len do by A177, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A183, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then A184: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A185: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A37, Th37;
A186: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
p in L~ (Lower_Seq (C,n)) by A53, A177;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A144, A176, XBOOLE_0:def_4;
then A187: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A188: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A189: 1 + 1 <= len (Lower_Seq (C,n)) by A23, XXREAL_0:2;
then A190: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
consider jj2 being Element of NAT such that
A191: 1 <= jj2 and
A192: jj2 <= width (Gauge (C,n)) and
A193: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A194: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A195: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A191, A192, MATRIX_1:36;
A196: 1 <= Index (p,do) by A177, JORDAN3:8;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A197: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A189, SPPOL_2:9;
A198: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A138, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A199: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A200: [ii,jj] in Indices (Gauge (C,n)) and
A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A202: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A85, A179, A186, A198, FINSEQ_6:92, JORDAN1I:23;
A203: (jj + 1) + 1 <> jj ;
A204: 1 <= jj by A200, MATRIX_1:38;
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A179, A198, FINSEQ_6:92;
then A205: ii = len (Gauge (C,n)) by A179, A199, A201, A193, A195, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A194, XREAL_1:9;
then A206: ii - 1 >= 1 by XXREAL_0:2;
then A207: 1 <= ii -' 1 by XREAL_0:def_2;
A208: jj <= width (Gauge (C,n)) by A200, MATRIX_1:38;
then A209: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A16, A204, JORDAN1A:71;
A210: jj + 1 <= width (Gauge (C,n)) by A199, MATRIX_1:38;
ii + 1 <> ii ;
then A211: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A85, A186, A188, A199, A200, A201, A202, A203, GOBOARD5:def_6;
A212: ii <= len (Gauge (C,n)) by A200, MATRIX_1:38;
A213: 1 <= ii by A200, MATRIX_1:38;
A214: ii <= len (Gauge (C,n)) by A199, MATRIX_1:38;
A215: 1 <= jj + 1 by A199, MATRIX_1:38;
then A216: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A16, A210, JORDAN1A:71;
A217: 1 <= ii by A199, MATRIX_1:38;
then A218: (ii -' 1) + 1 = ii by XREAL_1:235;
then A219: ii -' 1 < len (Gauge (C,n)) by A214, NAT_1:13;
then A220: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A215, A210, A207, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A217, A214, A215, A210, GOBOARD5:1 ;
A221: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
then A222: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= (E-max C) `2 by A214, A210, A204, A211, A218, A206, JORDAN9:17;
A223: (E-max C) `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A221, A214, A210, A204, A211, A218, A206, JORDAN9:17;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A204, A208, A207, A219, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A213, A212, A204, A208, GOBOARD5:1 ;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A187, A178, A201, A202, A205, A222, A223, A220, A209, A216, GOBOARD7:7;
then A224: p in LSeg ((Lower_Seq (C,n)),1) by A85, A197, A186, TOPREAL1:def_3;
A225: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A29, A37, FINSEQ_4:19;
then A226: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A225, XXREAL_0:1;
A227: (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A32, A37, Th56;
0 + (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by A184, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by A227;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A228: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A229: p in LSeg (do,(Index (p,do))) by A177, JORDAN3:9;
1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
then A230: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by A226, A196, A228, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))) by A9, JORDAN3:8;
then A231: 1 + 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A227, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A196, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A232: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A232, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence contradiction by A224, A229, A185, A230, XBOOLE_0:3; ::_thesis: verum
end;
supposeA233: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) ;
then A234: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) = 2 by A196, A231, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A23, A233, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A224, A229, A185, A230, XBOOLE_0:def_4;
then A235: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A236: p in rng (Lower_Seq (C,n)) by A190, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A190, A235, FINSEQ_5:41;
then p = (Gauge (C,n)) * (i,j) by A37, A234, A236, FINSEQ_5:9;
then ((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n))) by A235, JORDAN1G:32;
then ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A3, A12, A16, JORDAN1A:71;
hence contradiction by A2, A14, A67, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A237: W is_a_component_of (L~ godo) ` and
A238: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A238, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A237, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A239: east_halfline (E-max C) c= UBD (L~ godo) by A238, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A239;
then E-max C in LeftComp godo by GOBRD14:36;
then Lower_Arc C meets L~ godo by A130, A131, A132, A140, A151, Th36;
then A240: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A141, XBOOLE_1:70;
A241: Lower_Arc C c= C by JORDAN6:61;
percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A240, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A241, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A81, A148; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction
then Lower_Arc C meets L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A241, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
theorem Th59: :: JORDAN1J:59
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C )
set Ga = Gauge (C,n);
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
set UA = Upper_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gij = (Gauge (C,n)) * (i,j);
set Gik = (Gauge (C,n)) * (i,k);
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} and
A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} and
A8: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) misses Upper_Arc C ; ::_thesis: contradiction
(Gauge (C,n)) * (i,j) in {((Gauge (C,n)) * (i,j))} by TARSKI:def_1;
then A9: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:def_4;
(Gauge (C,n)) * (i,k) in {((Gauge (C,n)) * (i,k))} by TARSKI:def_1;
then A10: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:def_4;
then A11: j <> k by A1, A2, A3, A5, A9, Th57;
A12: j <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
A13: 1 <= k by A3, A4, XXREAL_0:2;
A14: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A12, MATRIX_1:36;
A15: [i,k] in Indices (Gauge (C,n)) by A1, A2, A5, A13, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)));
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)));
A16: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A17: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then 1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A18: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A19: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A5, A13, A16, JORDAN1A:73 ;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A20: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A21: [1,k] in Indices (Gauge (C,n)) by A5, A13, MATRIX_1:36;
then A22: (Gauge (C,n)) * (i,k) <> (Upper_Seq (C,n)) . 1 by A1, A15, A18, A19, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
A23: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A24: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A25: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A24, FINSEQ_3:25;
then A26: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
A27: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A5, A13, A16, JORDAN1A:73 ;
A28: [i,j] in Indices (Gauge (C,n)) by A1, A2, A3, A12, MATRIX_1:36;
then A29: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A21, A26, A27, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34;
A30: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A5, A13, A20, MATRIX_1:36;
A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A5, A13, A16, JORDAN1A:71 ;
then A32: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . 1 by A2, A28, A30, A31, JORDAN1G:7;
A33: len go >= 1 + 1 by TOPREAL1:def_8;
A34: (Gauge (C,n)) * (i,k) in rng (Upper_Seq (C,n)) by A1, A2, A5, A10, A13, Th40, JORDAN1G:4;
then A35: go is_sequence_on Gauge (C,n) by Th38, JORDAN1G:4;
A36: len do >= 1 + 1 by TOPREAL1:def_8;
A37: (Gauge (C,n)) * (i,j) in rng (Lower_Seq (C,n)) by A1, A2, A3, A9, A12, Th40, JORDAN1G:5;
then A38: do is_sequence_on Gauge (C,n) by Th39, JORDAN1G:5;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5;
A39: len go > 1 by A33, NAT_1:13;
then A40: len go in dom go by FINSEQ_3:25;
then A41: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,k) by A10, JORDAN3:24 ;
len do >= 1 by A36, XXREAL_0:2;
then 1 in dom do by FINSEQ_3:25;
then A42: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,j) by A9, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A40, FINSEQ_3:26;
A43: m + 1 = len go ;
then A44: (len go) -' 1 = m by NAT_D:34;
A45: LSeg (go,m) c= L~ go by TOPREAL3:19;
A46: L~ go c= L~ (Upper_Seq (C,n)) by A10, JORDAN3:41;
then LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A45, XBOOLE_1:1;
then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,k))} by A6, XBOOLE_1:26;
m >= 1 by A33, XREAL_1:19;
then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i,k))) by A41, A43, TOPREAL1:def_3;
{((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
A49: (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
assume x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A50: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1;
(Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
hence x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A50, A49, XBOOLE_0:def_4; ::_thesis: verum
end;
then A51: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,k))} by A47, XBOOLE_0:def_10;
A52: LSeg (do,1) c= L~ do by TOPREAL3:19;
A53: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42;
then LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A52, XBOOLE_1:1;
then A54: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,j))} by A7, XBOOLE_1:26;
A55: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i,j)),(do /. (1 + 1))) by A36, A42, TOPREAL1:def_3;
{((Gauge (C,n)) * (i,j))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
A56: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
assume x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A57: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1;
(Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68;
hence x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A57, A56, XBOOLE_0:def_4; ::_thesis: verum
end;
then A58: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A54, XBOOLE_0:def_10;
A59: go /. 1 = (Upper_Seq (C,n)) /. 1 by A10, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A60: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A9, Th35 ;
A61: rng go c= L~ go by A33, SPPOL_2:18;
A62: rng do c= L~ do by A36, SPPOL_2:18;
A63: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A64: x = go /. 1 by TARSKI:def_1;
then A65: x in rng go by FINSEQ_6:42;
x in rng do by A60, A64, REVROT_1:3;
hence x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A3, A12, A20, MATRIX_1:36;
(L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A68: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A69: x in L~ do by XBOOLE_0:def_4;
A70: now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A71: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,j) by A9, A66, A69, JORDAN1E:7;
((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A3, A12, A16, JORDAN1A:71;
then (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A14, A67, A71, JORDAN1G:7;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
x in L~ go by A68, XBOOLE_0:def_4;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A46, A53, A69, XBOOLE_0:def_4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
hence x in {(go /. 1)} by A59, A70, TARSKI:def_1; ::_thesis: verum
end;
then A72: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def_10;
set W2 = go /. 2;
A73: 2 in dom go by A33, FINSEQ_3:25;
A74: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume ((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A5, A13, A16, JORDAN1A:73;
hence contradiction by A1, A15, A21, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)))) by A34, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n))) by A34, FINSEQ_4:21, FINSEQ_6:116 ;
then A75: go /. 2 = (Upper_Seq (C,n)) /. 2 by A73, FINSEQ_4:70;
set pion = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>;
A76: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) )
then n in Seg 2 by FINSEQ_1:89;
then ( n = 1 or n = 2 ) by FINSEQ_1:2, TARSKI:def_2;
hence ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A14, A15, FINSEQ_4:17; ::_thesis: verum
end;
A77: (Gauge (C,n)) * (i,k) <> (Gauge (C,n)) * (i,j) by A11, A14, A15, GOBOARD1:5;
A78: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A5, A13, GOBOARD5:2
.= ((Gauge (C,n)) * (i,j)) `1 by A1, A2, A3, A12, GOBOARD5:2 ;
then LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) is vertical by SPPOL_1:16;
then <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> is being_S-Seq by A77, JORDAN1B:7;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A79: pion1 is_sequence_on Gauge (C,n) and
A80: pion1 is being_S-Seq and
A81: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = L~ pion1 and
A82: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 1 = pion1 /. 1 and
A83: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = pion1 /. (len pion1) and
A84: len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> <= len pion1 by A76, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A80;
set godo = (go ^' pion1) ^' do;
A85: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A86: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
len (go ^' pion1) >= len go by TOPREAL8:7;
then A87: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2;
then A88: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A89: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A90: 1 + 1 <= len ((go ^' pion1) ^' do) by A87, XXREAL_0:2;
A91: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A92: go /. (len go) = pion1 /. 1 by A41, A82, FINSEQ_4:17;
then A93: go ^' pion1 is_sequence_on Gauge (C,n) by A35, A79, TOPREAL8:12;
A94: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) by A83, GRAPH_2:54
.= <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
then A95: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A38, A93, TOPREAL8:12;
LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A81, TOPREAL3:19;
then LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A96: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i,k))} by A44, A51, XBOOLE_1:27;
A97: len pion1 >= 1 + 1 by A84, FINSEQ_1:44;
{((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A98: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1;
A99: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
(Gauge (C,n)) * (i,k) in LSeg (pion1,1) by A41, A92, A97, TOPREAL1:21;
hence x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A98, A99, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A96, XBOOLE_0:def_10;
then A100: go ^' pion1 is unfolded by A92, TOPREAL8:34;
len pion1 >= 2 + 0 by A84, FINSEQ_1:44;
then A101: (len pion1) - 2 >= 0 by XREAL_1:19;
((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A101, XREAL_0:def_2 ;
then A102: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A103: (len pion1) - 1 >= 1 by A97, XREAL_1:19;
then A104: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A105: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A101, XREAL_0:def_2
.= (len pion1) -' 1 by A103, XREAL_0:def_2 ;
((len pion1) - 1) + 1 <= len pion1 ;
then A106: (len pion1) -' 1 < len pion1 by A104, NAT_1:13;
LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A81, TOPREAL3:19;
then LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A107: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i,j))} by A58, XBOOLE_1:27;
{((Gauge (C,n)) * (i,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A108: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by A83, A104, FINSEQ_1:44
.= (Gauge (C,n)) * (i,j) by FINSEQ_4:17 ;
then A109: (Gauge (C,n)) * (i,j) in LSeg (pion1,((len pion1) -' 1)) by A103, A104, TOPREAL1:21;
(Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68;
hence x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A108, A109, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A107, XBOOLE_0:def_10;
then A110: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A92, A94, A105, A106, TOPREAL8:31;
A111: not go ^' pion1 is trivial by A87, NAT_D:60;
A112: rng pion1 c= L~ pion1 by A97, SPPOL_2:18;
A113: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A114: x = pion1 /. 1 by TARSKI:def_1;
then A115: x in rng pion1 by FINSEQ_6:42;
x in rng go by A92, A114, REVROT_1:3;
hence x in (L~ go) /\ (L~ pion1) by A61, A112, A115, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A116: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A117: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ go by A116, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A117, XBOOLE_0:def_4;
hence x in {(pion1 /. 1)} by A6, A41, A81, A92, SPPOL_2:21; ::_thesis: verum
end;
then A118: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A113, XBOOLE_0:def_10;
then A119: go ^' pion1 is s.n.c. by A92, Th54;
(rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A112, A118, XBOOLE_1:27;
then A120: go ^' pion1 is one-to-one by Th55;
A121: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
A122: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A123: x = pion1 /. (len pion1) by TARSKI:def_1;
then A124: x in rng pion1 by REVROT_1:3;
x in rng do by A83, A121, A123, FINSEQ_6:42;
hence x in (L~ do) /\ (L~ pion1) by A62, A112, A124, XBOOLE_0:def_4; ::_thesis: verum
end;
(L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A125: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A126: x in L~ pion1 by XBOOLE_0:def_4;
x in L~ do by A125, XBOOLE_0:def_4;
then x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A126, XBOOLE_0:def_4;
hence x in {(pion1 /. (len pion1))} by A7, A42, A81, A83, A121, SPPOL_2:21; ::_thesis: verum
end;
then A127: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A122, XBOOLE_0:def_10;
A128: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A92, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A72, A83, A121, A127, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
do /. (len do) = (go ^' pion1) /. 1 by A60, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A90, A94, A95, A100, A102, A110, A111, A119, A120, A128, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A129: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def_8;
then A130: Upper_Arc C is connected by JORDAN6:10;
A131: W-min C in Upper_Arc C by A129, TOPREAL1:1;
A132: E-max C in Upper_Arc C by A129, TOPREAL1:1;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A133: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A134: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, SPRECT_5:22;
then (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:23, XXREAL_0:2;
then (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:24, XXREAL_0:2;
then A135: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A133, A134, SPRECT_5:25, XXREAL_0:2;
A136: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_.._(Upper_Seq_(C,n))_<=_1
assume A137: ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) >= 1 by A34, FINSEQ_4:21;
then ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) = 1 by A137, XXREAL_0:1;
then (Gauge (C,n)) * (i,k) = (Upper_Seq (C,n)) /. 1 by A34, FINSEQ_5:38;
hence contradiction by A18, A22, JORDAN1F:5; ::_thesis: verum
end;
A138: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A139: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A140: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A90, A95, JORDAN9:27;
A141: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A94, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A92, TOPREAL8:35 ;
A142: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A143: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A144: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A142, XBOOLE_1:7;
A145: L~ go c= L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1;
A146: L~ do c= L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1;
A147: W-min C in C by SPRECT_1:13;
A148: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
A149: now__::_thesis:_not_W-min_C_in_L~_godo
assume W-min C in L~ godo ; ::_thesis: contradiction
then A150: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A141, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A150, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A145, A147, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A81, A131, A148, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then C meets L~ (Cage (C,n)) by A146, A147, XBOOLE_0:3;
hence contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A86, JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A135, A139, Th53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)))),1,(Gauge (C,n))) by A34, A91, A136, Th52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A93, Th51
.= right_cell (godo,1,(Gauge (C,n))) by A88, A95, Th51 ;
then W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A151: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A149, XBOOLE_0:def_5;
A152: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A59, GRAPH_2:53 ;
A153: len (Upper_Seq (C,n)) >= 2 by A17, XXREAL_0:2;
A154: godo /. 2 = (go ^' pion1) /. 2 by A87, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A33, A75, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A153, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A155: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42;
then W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A61, XBOOLE_0:def_3;
then A156: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A145, A146, A155, Th21, XBOOLE_1:8;
A157: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A158: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
W-bound (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = ((Gauge (C,n)) * (i,k)) `1 by A78, SPRECT_1:54;
then A159: W-bound (L~ pion1) = ((Gauge (C,n)) * (i,k)) `1 by A81, SPPOL_2:21;
((Gauge (C,n)) * (i,k)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A143, PSCOMP_1:24;
then ((Gauge (C,n)) * (i,k)) `1 > W-bound (L~ (Cage (C,n))) by A74, XXREAL_0:1;
then W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A155, A156, A157, A158, A159, Th33;
then A160: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A141, A156, XBOOLE_1:4;
A161: rng godo c= L~ godo by A87, A89, SPPOL_2:18, XXREAL_0:2;
2 in dom godo by A90, FINSEQ_3:25;
then A162: godo /. 2 in rng godo by PARTFUN2:2;
godo /. 2 in W-most (L~ (Cage (C,n))) by A154, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A160, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then godo /. 2 in W-most (L~ godo) by A161, A162, SPRECT_2:12;
then (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A152, A160, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A163: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A164: east_halfline (E-max C) misses L~ go
proof
assume east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set such that
A165: p in east_halfline (E-max C) and
A166: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A165;
p in L~ (Upper_Seq (C,n)) by A46, A166;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A143, A165, XBOOLE_0:def_4;
then A167: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A168: p = E-max (L~ (Cage (C,n))) by A46, A166, Th46;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,k) by A10, A163, A166, Th43;
then ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A5, A13, A16, A167, A168, JORDAN1A:71;
hence contradiction by A2, A15, A30, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A169: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A141, XBOOLE_1:70;
percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A169, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence contradiction by A164; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set such that
A170: p in east_halfline (E-max C) and
A171: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A170;
A172: p `2 = (E-max C) `2 by A170, TOPREAL1:def_11;
i + 1 <= len (Gauge (C,n)) by A2, NAT_1:13;
then (i + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A173: i <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
A174: (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
p `1 = ((Gauge (C,n)) * (i,k)) `1 by A78, A81, A148, A171, GOBOARD7:5;
then p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A1, A5, A13, A16, A20, A173, A174, JORDAN1A:18;
then p `1 <= E-bound C by A20, JORDAN8:12;
then A175: p `1 <= (E-max C) `1 by EUCLID:52;
p `1 >= (E-max C) `1 by A170, TOPREAL1:def_11;
then p `1 = (E-max C) `1 by A175, XXREAL_0:1;
then p = E-max C by A172, TOPREAL3:6;
hence contradiction by A8, A81, A132, A148, A171, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set such that
A176: p in east_halfline (E-max C) and
A177: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A176;
A178: (E-max C) `2 = p `2 by A176, TOPREAL1:def_11;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A179: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
consider t being Nat such that
A180: t in dom (Lower_Seq (C,n)) and
A181: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i,j) by A37, FINSEQ_2:10;
1 <= t by A180, FINSEQ_3:25;
then A182: 1 < t by A32, A181, XXREAL_0:1;
t <= len (Lower_Seq (C,n)) by A180, FINSEQ_3:25;
then (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = t by A181, A182, JORDAN3:12;
then A183: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A9, A181, JORDAN3:26;
Index (p,do) < len do by A177, JORDAN3:8;
then Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A183, XREAL_0:def_2;
then (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then A184: Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
A185: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A37, Th37;
A186: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
p in L~ (Lower_Seq (C,n)) by A53, A177;
then p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A144, A176, XBOOLE_0:def_4;
then A187: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A188: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A189: 1 + 1 <= len (Lower_Seq (C,n)) by A23, XXREAL_0:2;
then A190: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
consider jj2 being Element of NAT such that
A191: 1 <= jj2 and
A192: jj2 <= width (Gauge (C,n)) and
A193: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A194: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A195: [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A191, A192, MATRIX_1:36;
A196: 1 <= Index (p,do) by A177, JORDAN3:8;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A197: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A189, SPPOL_2:9;
A198: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A138, REVROT_1:34;
then consider ii, jj being Element of NAT such that
A199: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A200: [ii,jj] in Indices (Gauge (C,n)) and
A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A202: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A85, A179, A186, A198, FINSEQ_6:92, JORDAN1I:23;
A203: (jj + 1) + 1 <> jj ;
A204: 1 <= jj by A200, MATRIX_1:38;
(Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A179, A198, FINSEQ_6:92;
then A205: ii = len (Gauge (C,n)) by A179, A199, A201, A193, A195, GOBOARD1:5;
then ii - 1 >= 4 - 1 by A194, XREAL_1:9;
then A206: ii - 1 >= 1 by XXREAL_0:2;
then A207: 1 <= ii -' 1 by XREAL_0:def_2;
A208: jj <= width (Gauge (C,n)) by A200, MATRIX_1:38;
then A209: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A16, A204, JORDAN1A:71;
A210: jj + 1 <= width (Gauge (C,n)) by A199, MATRIX_1:38;
ii + 1 <> ii ;
then A211: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A85, A186, A188, A199, A200, A201, A202, A203, GOBOARD5:def_6;
A212: ii <= len (Gauge (C,n)) by A200, MATRIX_1:38;
A213: 1 <= ii by A200, MATRIX_1:38;
A214: ii <= len (Gauge (C,n)) by A199, MATRIX_1:38;
A215: 1 <= jj + 1 by A199, MATRIX_1:38;
then A216: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A16, A210, JORDAN1A:71;
A217: 1 <= ii by A199, MATRIX_1:38;
then A218: (ii -' 1) + 1 = ii by XREAL_1:235;
then A219: ii -' 1 < len (Gauge (C,n)) by A214, NAT_1:13;
then A220: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A215, A210, A207, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A217, A214, A215, A210, GOBOARD5:1 ;
A221: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
then A222: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= (E-max C) `2 by A214, A210, A204, A211, A218, A206, JORDAN9:17;
A223: (E-max C) `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A221, A214, A210, A204, A211, A218, A206, JORDAN9:17;
((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A204, A208, A207, A219, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A213, A212, A204, A208, GOBOARD5:1 ;
then p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A187, A178, A201, A202, A205, A222, A223, A220, A209, A216, GOBOARD7:7;
then A224: p in LSeg ((Lower_Seq (C,n)),1) by A85, A197, A186, TOPREAL1:def_3;
A225: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A29, A37, FINSEQ_4:19;
then A226: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A225, XXREAL_0:1;
A227: (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A32, A37, Th56;
0 + (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8;
then (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
then Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by A184, XREAL_0:def_2;
then Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by A227;
then Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then A228: Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
A229: p in LSeg (do,(Index (p,do))) by A177, JORDAN3:9;
1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
then A230: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by A226, A196, A228, JORDAN4:19;
1 <= Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))) by A9, JORDAN3:8;
then A231: 1 + 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A227, XREAL_1:7;
then (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A196, XREAL_1:7;
then ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A232: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A232, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence contradiction by A224, A229, A185, A230, XBOOLE_0:3; ::_thesis: verum
end;
supposeA233: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 by XREAL_0:def_2;
then (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) ;
then A234: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) = 2 by A196, A231, JORDAN1E:6;
(LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A23, A233, TOPREAL1:def_6;
then p in {((Lower_Seq (C,n)) /. 2)} by A224, A229, A185, A230, XBOOLE_0:def_4;
then A235: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A236: p in rng (Lower_Seq (C,n)) by A190, PARTFUN2:2;
p .. (Lower_Seq (C,n)) = 2 by A190, A235, FINSEQ_5:41;
then p = (Gauge (C,n)) * (i,j) by A37, A234, A236, FINSEQ_5:9;
then ((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n))) by A235, JORDAN1G:32;
then ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A3, A12, A16, JORDAN1A:71;
hence contradiction by A2, A14, A67, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A237: W is_a_component_of (L~ godo) ` and
A238: east_halfline (E-max C) c= W by GOBOARD9:3;
not W is bounded by A238, JORDAN2C:121, RLTOPSP1:42;
then W is_outside_component_of L~ godo by A237, JORDAN2C:def_3;
then W c= UBD (L~ godo) by JORDAN2C:23;
then A239: east_halfline (E-max C) c= UBD (L~ godo) by A238, XBOOLE_1:1;
E-max C in east_halfline (E-max C) by TOPREAL1:38;
then E-max C in UBD (L~ godo) by A239;
then E-max C in LeftComp godo by GOBRD14:36;
then Upper_Arc C meets L~ godo by A130, A131, A132, A140, A151, Th36;
then A240: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A141, XBOOLE_1:70;
A241: Upper_Arc C c= C by JORDAN6:61;
percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A240, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A46, A143, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A241, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence contradiction by A8, A81, A148; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction
then Upper_Arc C meets L~ (Cage (C,n)) by A53, A144, XBOOLE_1:1, XBOOLE_1:63;
hence contradiction by A241, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
theorem Th60: :: JORDAN1J:60
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: n > 0 and
A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} and
A8: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th58; ::_thesis: verum
end;
theorem Th61: :: JORDAN1J:61
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: n > 0 and
A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} and
A8: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th59; ::_thesis: verum
end;
theorem :: JORDAN1J:62
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds
LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1))))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds
LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1))))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds
LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1))))
let j be Element of NAT ; ::_thesis: ( (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) implies LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) )
assume that
A1: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Upper_Arc (L~ (Cage (C,(n + 1)))) and
A2: 1 <= j and
A3: j <= width (Gauge (C,(n + 1))) ; ::_thesis: LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1))))
set in1 = Center (Gauge (C,(n + 1)));
A4: 1 <= Center (Gauge (C,(n + 1))) by JORDAN1B:11;
A5: Upper_Arc (L~ (Cage (C,(n + 1)))) c= L~ (Cage (C,(n + 1))) by JORDAN6:61;
A6: Center (Gauge (C,(n + 1))) <= len (Gauge (C,(n + 1))) by JORDAN1B:13;
n + 1 >= 0 + 1 by NAT_1:11;
then LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) c= LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) by A2, A3, JORDAN1A:45;
hence LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),1)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) by A1, A2, A3, A4, A6, A5, JORDAN1G:57, XBOOLE_1:63; ::_thesis: verum
end;
theorem :: JORDAN1J:63
for n being Element of NAT
for C being Simple_closed_curve
for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C
let j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} implies LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C )
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= width (Gauge (C,(n + 1))) and
A4: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} and
A5: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 2 by XXREAL_0:2;
then A7: 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
len (Gauge (C,(n + 1))) >= 3 by A6, XXREAL_0:2;
hence LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, Th60, JORDAN1B:15; ::_thesis: verum
end;
theorem :: JORDAN1J:64
for n being Element of NAT
for C being Simple_closed_curve
for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for j, k being Element of NAT st 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} holds
LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
let j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} & (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} implies LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C )
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= width (Gauge (C,(n + 1))) and
A4: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Upper_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))} and
A5: (LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k)))) /\ (Lower_Arc (L~ (Cage (C,(n + 1))))) = {((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))} ; ::_thesis: LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C
A6: len (Gauge (C,(n + 1))) >= 4 by JORDAN8:10;
then len (Gauge (C,(n + 1))) >= 2 by XXREAL_0:2;
then A7: 1 < Center (Gauge (C,(n + 1))) by JORDAN1B:14;
len (Gauge (C,(n + 1))) >= 3 by A6, XXREAL_0:2;
hence LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j)),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),k))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, Th61, JORDAN1B:15; ::_thesis: verum
end;
theorem :: JORDAN1J:65
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & ( W-min Y in X or W-max Y in X ) holds
W-bound X = W-bound Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & ( W-min Y in X or W-max Y in X ) implies W-bound X = W-bound Y )
assume that
A1: X c= Y and
A2: ( W-min Y in X or W-max Y in X ) ; ::_thesis: W-bound X = W-bound Y
A3: (W-max X) `1 = W-bound X by EUCLID:52;
A4: (W-max Y) `1 = W-bound Y by EUCLID:52;
A5: (W-min Y) `1 = W-bound Y by EUCLID:52;
(W-min X) `1 = W-bound X by EUCLID:52;
hence W-bound X = W-bound Y by A1, A2, A3, A5, A4, Th21, Th22; ::_thesis: verum
end;
theorem :: JORDAN1J:66
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & ( E-min Y in X or E-max Y in X ) holds
E-bound X = E-bound Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & ( E-min Y in X or E-max Y in X ) implies E-bound X = E-bound Y )
assume that
A1: X c= Y and
A2: ( E-min Y in X or E-max Y in X ) ; ::_thesis: E-bound X = E-bound Y
A3: (E-max X) `1 = E-bound X by EUCLID:52;
A4: (E-max Y) `1 = E-bound Y by EUCLID:52;
A5: (E-min Y) `1 = E-bound Y by EUCLID:52;
(E-min X) `1 = E-bound X by EUCLID:52;
hence E-bound X = E-bound Y by A1, A2, A3, A5, A4, Th17, Th18; ::_thesis: verum
end;
theorem :: JORDAN1J:67
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & ( N-min Y in X or N-max Y in X ) holds
N-bound X = N-bound Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & ( N-min Y in X or N-max Y in X ) implies N-bound X = N-bound Y )
assume that
A1: X c= Y and
A2: ( N-min Y in X or N-max Y in X ) ; ::_thesis: N-bound X = N-bound Y
A3: (N-max X) `2 = N-bound X by EUCLID:52;
A4: (N-max Y) `2 = N-bound Y by EUCLID:52;
A5: (N-min Y) `2 = N-bound Y by EUCLID:52;
(N-min X) `2 = N-bound X by EUCLID:52;
hence N-bound X = N-bound Y by A1, A2, A3, A5, A4, Th15, Th16; ::_thesis: verum
end;
theorem :: JORDAN1J:68
for X, Y being non empty compact Subset of (TOP-REAL 2) st X c= Y & ( S-min Y in X or S-max Y in X ) holds
S-bound X = S-bound Y
proof
let X, Y be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( X c= Y & ( S-min Y in X or S-max Y in X ) implies S-bound X = S-bound Y )
assume that
A1: X c= Y and
A2: ( S-min Y in X or S-max Y in X ) ; ::_thesis: S-bound X = S-bound Y
A3: (S-max X) `2 = S-bound X by EUCLID:52;
A4: (S-max Y) `2 = S-bound Y by EUCLID:52;
A5: (S-min Y) `2 = S-bound Y by EUCLID:52;
(S-min X) `2 = S-bound X by EUCLID:52;
hence S-bound X = S-bound Y by A1, A2, A3, A5, A4, Th19, Th20; ::_thesis: verum
end;