let G be non empty TopSpace; :: thesis: for w1, w2, w3 being Point of G
for h1, h2 being Function of I[01] ,G st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01] ,G st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) )
let w1, w2, w3 be Point of G; :: thesis: for h1, h2 being Function of I[01] ,G st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01] ,G st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) )
let h1, h2 be Function of I[01] ,G; :: thesis: ( h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 implies ex h3 being Function of I[01] ,G st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) ) )
assume that
A1: h1 is continuous and
A2: w1 = h1 . 0 and
A3: w2 = h1 . 1 and
A4: h2 is continuous and
A5: w2 = h2 . 0 and
A6: w3 = h2 . 1 ; :: thesis: ex h3 being Function of I[01] ,G st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) )
w2,w3 are_connected by A4, A5, A6, Def1;
then reconsider g2 = h2 as Path of w2,w3 by A4, A5, A6, Def2;
w1,w2 are_connected by A1, A2, A3, Def1;
then reconsider g1 = h1 as Path of w1,w2 by A1, A2, A3, Def2;
set P1 = g1;
set P2 = g2;
set p1 = w1;
set p3 = w3;
ex P0 being Path of w1,w3 st
( P0 is continuous & P0 . 0 = w1 & P0 . 1 = w3 & ( for t being Point of I[01]
for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies P0 . t = g1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies P0 . t = g2 . ((2 * t9) - 1) ) ) ) )
proof
1 / 2 in { r where r is Real : ( 0 <= r & r <= 1 ) } ;
then reconsider pol = 1 / 2 as Point of I[01] by BORSUK_1:83, RCOMP_1:def 1;
reconsider T1 = Closed-Interval-TSpace 0 ,(1 / 2), T2 = Closed-Interval-TSpace (1 / 2),1 as SubSpace of I[01] by TOPMETR:27, TREAL_1:6;
set e2 = P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) );
set e1 = P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) );
set E1 = g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ));
set E2 = g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ));
set f = (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) +* (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )));
A7: dom (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )) = the carrier of (Closed-Interval-TSpace 0 ,(1 / 2)) by FUNCT_2:def 1
.= [.0 ,(1 / 2).] by TOPMETR:25 ;
A8: dom (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )) = the carrier of (Closed-Interval-TSpace (1 / 2),1) by FUNCT_2:def 1
.= [.(1 / 2),1.] by TOPMETR:25 ;
reconsider gg = g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )) as Function of T2,G by TOPMETR:27;
reconsider ff = g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )) as Function of T1,G by TOPMETR:27;
A9: for t9 being Real st 1 / 2 <= t9 & t9 <= 1 holds
(g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) . t9 = g2 . ((2 * t9) - 1)
proof
reconsider r1 = (#) 0 ,1, r2 = 0 ,1 (#) as Real by TREAL_1:8;
dom (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )) = the carrier of (Closed-Interval-TSpace (1 / 2),1) by FUNCT_2:def 1;
then A10: dom (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )) = [.(1 / 2),1.] by TOPMETR:25
.= { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } by RCOMP_1:def 1 ;
let t9 be Real; :: thesis: ( 1 / 2 <= t9 & t9 <= 1 implies (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) . t9 = g2 . ((2 * t9) - 1) )
assume ( 1 / 2 <= t9 & t9 <= 1 ) ; :: thesis: (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) . t9 = g2 . ((2 * t9) - 1)
then A11: t9 in dom (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )) by A10;
then reconsider s = t9 as Point of (Closed-Interval-TSpace (1 / 2),1) ;
(P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )) . s = (((r2 - r1) / (1 - (1 / 2))) * t9) + (((1 * r1) - ((1 / 2) * r2)) / (1 - (1 / 2))) by TREAL_1:14
.= (2 * t9) - 1 by TREAL_1:8 ;
hence (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) . t9 = g2 . ((2 * t9) - 1) by A11, FUNCT_1:23; :: thesis: verum
end;
A12: for t9 being Real st 0 <= t9 & t9 <= 1 / 2 holds
(g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) . t9 = g1 . (2 * t9)
proof
reconsider r1 = (#) 0 ,1, r2 = 0 ,1 (#) as Real by TREAL_1:8;
dom (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )) = the carrier of (Closed-Interval-TSpace 0 ,(1 / 2)) by FUNCT_2:def 1;
then A13: dom (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )) = [.0 ,(1 / 2).] by TOPMETR:25
.= { r where r is Real : ( 0 <= r & r <= 1 / 2 ) } by RCOMP_1:def 1 ;
let t9 be Real; :: thesis: ( 0 <= t9 & t9 <= 1 / 2 implies (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) . t9 = g1 . (2 * t9) )
assume ( 0 <= t9 & t9 <= 1 / 2 ) ; :: thesis: (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) . t9 = g1 . (2 * t9)
then A14: t9 in dom (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )) by A13;
then reconsider s = t9 as Point of (Closed-Interval-TSpace 0 ,(1 / 2)) ;
(P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )) . s = (((r2 - r1) / ((1 / 2) - 0 )) * t9) + ((((1 / 2) * r1) - (0 * r2)) / ((1 / 2) - 0 )) by TREAL_1:14
.= 2 * t9 by TREAL_1:8 ;
hence (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) . t9 = g1 . (2 * t9) by A14, FUNCT_1:23; :: thesis: verum
end;
then A15: ff . (1 / 2) = g2 . ((2 * (1 / 2)) - 1) by A3, A5
.= gg . pol by A9 ;
( [#] T1 = [.0 ,(1 / 2).] & [#] T2 = [.(1 / 2),1.] ) by TOPMETR:25;
then A16: ( ([#] T1) \/ ([#] T2) = [#] I[01] & ([#] T1) /\ ([#] T2) = {pol} ) by BORSUK_1:83, XXREAL_1:174, XXREAL_1:418;
A17: T2 is compact by HEINE:11;
dom g1 = the carrier of I[01] by FUNCT_2:def 1;
then A18: rng (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )) c= dom g1 by TOPMETR:27;
( dom g2 = the carrier of I[01] & rng (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )) c= the carrier of (Closed-Interval-TSpace 0 ,1) ) by FUNCT_2:def 1;
then A19: dom (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) = dom (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )) by RELAT_1:46, TOPMETR:27;
not 0 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) }
proof
assume 0 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } ; :: thesis: contradiction
then ex rr being Real st
( rr = 0 & 1 / 2 <= rr & rr <= 1 ) ;
hence contradiction ; :: thesis: verum
end;
then not 0 in dom (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) by A8, A19, RCOMP_1:def 1;
then A20: ((g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) +* (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )))) . 0 = (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) . 0 by FUNCT_4:12
.= g1 . (2 * 0 ) by A12
.= w1 by A2 ;
A21: dom ((g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) +* (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )))) = (dom (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )))) \/ (dom (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )))) by FUNCT_4:def 1
.= [.0 ,(1 / 2).] \/ [.(1 / 2),1.] by A7, A8, A18, A19, RELAT_1:46
.= the carrier of I[01] by BORSUK_1:83, XXREAL_1:174 ;
rng ((g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) +* (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )))) c= (rng (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) )))) \/ (rng (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )))) by FUNCT_4:18;
then A22: rng ((g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) +* (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )))) c= the carrier of G by XBOOLE_1:1;
A23: ( R^1 is T_2 & T1 is compact ) by HEINE:11, PCOMPS_1:38, TOPMETR:def 7;
reconsider f = (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) +* (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) as Function of I[01] ,G by A21, A22, FUNCT_2:def 1, RELSET_1:11;
( P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ) is continuous & P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ) is continuous ) by TREAL_1:15;
then reconsider f = f as continuous Function of I[01] ,G by A1, A4, A15, A16, A23, A17, COMPTS_1:29, TOPMETR:27;
1 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } ;
then 1 in dom (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) by A8, A19, RCOMP_1:def 1;
then A24: f . 1 = (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) . 1 by FUNCT_4:14
.= g2 . ((2 * 1) - 1) by A9
.= w3 by A6 ;
then w1,w3 are_connected by A20, Def1;
then reconsider f = f as Path of w1,w3 by A20, A24, Def2;
for t being Point of I[01]
for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies f . t = g1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies f . t = g2 . ((2 * t9) - 1) ) )
proof
let t be Point of I[01] ; :: thesis: for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies f . t = g1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies f . t = g2 . ((2 * t9) - 1) ) )
let t9 be Real; :: thesis: ( t = t9 implies ( ( 0 <= t9 & t9 <= 1 / 2 implies f . t = g1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies f . t = g2 . ((2 * t9) - 1) ) ) )
assume A25: t = t9 ; :: thesis: ( ( 0 <= t9 & t9 <= 1 / 2 implies f . t = g1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies f . t = g2 . ((2 * t9) - 1) ) )
thus ( 0 <= t9 & t9 <= 1 / 2 implies f . t = g1 . (2 * t9) ) :: thesis: ( 1 / 2 <= t9 & t9 <= 1 implies f . t = g2 . ((2 * t9) - 1) )
proof
assume A26: ( 0 <= t9 & t9 <= 1 / 2 ) ; :: thesis: f . t = g1 . (2 * t9)
then t9 in { r where r is Real : ( 0 <= r & r <= 1 / 2 ) } ;
then A27: t9 in [.0 ,(1 / 2).] by RCOMP_1:def 1;
per cases ( t9 <> 1 / 2 or t9 = 1 / 2 ) ;
suppose A28: t9 <> 1 / 2 ; :: thesis: f . t = g1 . (2 * t9)
not t9 in dom (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) )))
proof
assume t9 in dom (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) ; :: thesis: contradiction
then t9 in [.0 ,(1 / 2).] /\ [.(1 / 2),1.] by A8, A19, A27, XBOOLE_0:def 4;
then t9 in {(1 / 2)} by XXREAL_1:418;
hence contradiction by A28, TARSKI:def 1; :: thesis: verum
end;
then f . t = (g1 * (P[01] 0 ,(1 / 2),((#) 0 ,1),(0 ,1 (#) ))) . t by A25, FUNCT_4:12
.= g1 . (2 * t9) by A12, A25, A26 ;
hence f . t = g1 . (2 * t9) ; :: thesis: verum
end;
suppose A29: t9 = 1 / 2 ; :: thesis: f . t = g1 . (2 * t9)
1 / 2 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } ;
then 1 / 2 in [.(1 / 2),1.] by RCOMP_1:def 1;
then 1 / 2 in the carrier of (Closed-Interval-TSpace (1 / 2),1) by TOPMETR:25;
then t in dom (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) by A25, A29, FUNCT_2:def 1, TOPMETR:27;
then f . t = (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) . (1 / 2) by A25, A29, FUNCT_4:14
.= g1 . (2 * t9) by A12, A15, A29 ;
hence f . t = g1 . (2 * t9) ; :: thesis: verum
end;
end;
end;
thus ( 1 / 2 <= t9 & t9 <= 1 implies f . t = g2 . ((2 * t9) - 1) ) :: thesis: verum
proof
assume A30: ( 1 / 2 <= t9 & t9 <= 1 ) ; :: thesis: f . t = g2 . ((2 * t9) - 1)
then t9 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } ;
then t9 in [.(1 / 2),1.] by RCOMP_1:def 1;
then f . t = (g2 * (P[01] (1 / 2),1,((#) 0 ,1),(0 ,1 (#) ))) . t by A8, A19, A25, FUNCT_4:14
.= g2 . ((2 * t9) - 1) by A9, A25, A30 ;
hence f . t = g2 . ((2 * t9) - 1) ; :: thesis: verum
end;
end;
hence ex P0 being Path of w1,w3 st
( P0 is continuous & P0 . 0 = w1 & P0 . 1 = w3 & ( for t being Point of I[01]
for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies P0 . t = g1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies P0 . t = g2 . ((2 * t9) - 1) ) ) ) ) by A20, A24; :: thesis: verum
end;
then consider P0 being Path of w1,w3 such that
A31: ( P0 is continuous & P0 . 0 = w1 & P0 . 1 = w3 ) and
A32: for t being Point of I[01]
for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies P0 . t = g1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies P0 . t = g2 . ((2 * t9) - 1) ) ) ;
rng P0 c= (rng g1) \/ (rng g2)
proof
A33: dom g2 = the carrier of I[01] by FUNCT_2:def 1;
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng P0 or x in (rng g1) \/ (rng g2) )
A34: dom g1 = the carrier of I[01] by FUNCT_2:def 1;
assume x in rng P0 ; :: thesis: x in (rng g1) \/ (rng g2)
then consider z being set such that
A35: z in dom P0 and
A36: x = P0 . z by FUNCT_1:def 5;
dom P0 = the carrier of I[01] by FUNCT_2:def 1;
then reconsider r = z as Real by A35, BORSUK_1:83;
A37: 0 <= r by A35, BORSUK_1:83, XXREAL_1:1;
A38: r <= 1 by A35, BORSUK_1:83, XXREAL_1:1;
per cases ( r <= 1 / 2 or r > 1 / 2 ) ;
suppose A42: r > 1 / 2 ; :: thesis: x in (rng g1) \/ (rng g2)
2 * r <= 2 * 1 by A38, XREAL_1:66;
then 2 * r <= 1 + 1 ;
then A43: (2 * r) - 1 <= 1 by XREAL_1:22;
2 * (1 / 2) = 1 ;
then 0 + 1 <= 2 * r by A42, XREAL_1:66;
then 0 <= (2 * r) - 1 by XREAL_1:21;
then A44: (2 * r) - 1 in the carrier of I[01] by A43, BORSUK_1:83, XXREAL_1:1;
P0 . z = g2 . ((2 * r) - 1) by A32, A35, A38, A42;
then P0 . z in rng g2 by A33, A44, FUNCT_1:def 5;
hence x in (rng g1) \/ (rng g2) by A36, XBOOLE_0:def 3; :: thesis: verum
end;
end;
end;
hence ex h3 being Function of I[01] ,G st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) ) by A31; :: thesis: verum