let P, Q be Subset of (TOP-REAL 2); :: thesis: for p1, p2, q1 being Point of (TOP-REAL 2)
for f being Function of I[01],((TOP-REAL 2) | P)
for s1 being Real st q1 in P & f . s1 = q1 & f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 & 0 <= s1 & s1 <= 1 & ( for t being Real st 0 <= t & t < s1 holds
not f . t in Q ) holds
for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = q1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q
let p1, p2, q1 be Point of (TOP-REAL 2); :: thesis: for f being Function of I[01],((TOP-REAL 2) | P)
for s1 being Real st q1 in P & f . s1 = q1 & f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 & 0 <= s1 & s1 <= 1 & ( for t being Real st 0 <= t & t < s1 holds
not f . t in Q ) holds
for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = q1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q
let f be Function of I[01],((TOP-REAL 2) | P); :: thesis: for s1 being Real st q1 in P & f . s1 = q1 & f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 & 0 <= s1 & s1 <= 1 & ( for t being Real st 0 <= t & t < s1 holds
not f . t in Q ) holds
for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = q1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q
let s1 be Real; :: thesis: ( q1 in P & f . s1 = q1 & f is being_homeomorphism & f . 0 = p1 & f . 1 = p2 & 0 <= s1 & s1 <= 1 & ( for t being Real st 0 <= t & t < s1 holds
not f . t in Q ) implies for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = q1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q )
assume that
A1: q1 in P and
A2: f . s1 = q1 and
A3: f is being_homeomorphism and
A4: f . 0 = p1 and
A5: f . 1 = p2 and
A6: ( 0 <= s1 & s1 <= 1 ) and
A7: for t being Real st 0 <= t & t < s1 holds
not f . t in Q ; :: thesis: for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = q1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q
reconsider P9 = P as non empty Subset of (TOP-REAL 2) by A1;
let g be Function of I[01],((TOP-REAL 2) | P); :: thesis: for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = q1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 0 <= t & t < s2 holds
not g . t in Q
let s2 be Real; :: thesis: ( g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = q1 & 0 <= s2 & s2 <= 1 implies for t being Real st 0 <= t & t < s2 holds
not g . t in Q )
assume that
A8: g is being_homeomorphism and
A9: g . 0 = p1 and
A10: g . 1 = p2 and
A11: g . s2 = q1 and
A12: 0 <= s2 and
A13: s2 <= 1 ; :: thesis: for t being Real st 0 <= t & t < s2 holds
not g . t in Q
reconsider f = f, g = g as Function of I[01],((TOP-REAL 2) | P9) ;
A14: f is one-to-one by A3, TOPS_2:def 5;
A15: dom f = [#] I[01] by A3, TOPS_2:def 5;
then A16: 1 in dom f by BORSUK_1:43;
A17: rng f = [#] ((TOP-REAL 2) | P) by A3, TOPS_2:def 5;
then f is onto by FUNCT_2:def 3;
then A18: f " = f " by A14, TOPS_2:def 4;
A19: (f ") . p2 = 1 by A5, A16, A14, A18, FUNCT_1:32;
A20: 0 in dom f by A15, BORSUK_1:43;
A21: (f ") . p1 = 0 by A4, A20, A14, A18, FUNCT_1:32;
set fg = (f ") * g;
A22: f " is being_homeomorphism by A3, TOPS_2:56;
then (f ") * g is being_homeomorphism by A8, TOPS_2:57;
then A23: ( (f ") * g is continuous & (f ") * g is one-to-one ) by TOPS_2:def 5;
let t be Real; :: thesis: ( 0 <= t & t < s2 implies not g . t in Q )
assume that
A24: 0 <= t and
A25: t < s2 ; :: thesis: not g . t in Q
A26: t <= 1 by A13, A25, XXREAL_0:2;
then reconsider t1 = t, s29 = s2 as Point of I[01] by A12, A13, A24, BORSUK_1:43;
A27: t in the carrier of I[01] by A24, A26, BORSUK_1:43;
reconsider Ft = ((f ") * g) . t1 as Real by BORSUK_1:40;
A28: rng g = [#] ((TOP-REAL 2) | P) by A8, TOPS_2:def 5;
A29: dom g = [#] I[01] by A8, TOPS_2:def 5;
then 1 in dom g by BORSUK_1:43;
then A30: ((f ") * g) . 1 = 1 by A10, A19, FUNCT_1:13;
A31: s1 in dom f by A6, A15, BORSUK_1:43;
dom (f ") = [#] ((TOP-REAL 2) | P) by A22, TOPS_2:def 5;
then A32: dom ((f ") * g) = dom g by A28, RELAT_1:27;
0 in dom g by A29, BORSUK_1:43;
then A33: ((f ") * g) . 0 = 0 by A9, A21, FUNCT_1:13;
A34: 0 <= Ft

proof

per cases ( 0 < t or 0 = t ) by A24;

suppose 0 < t ; :: thesis: 0 <= Ft

hence 0 <= Ft by A23, A33, A30, BORSUK_1:def 14, JORDAN5A:15, TOPMETR:20; :: thesis: verum

end;

suppose 0 = t ; :: thesis: 0 <= Ft

hence 0 <= Ft by A9, A21, A29, FUNCT_1:13; :: thesis: verum

end;

end;

end;

f * ((f ") * g) = g * (f * (f ")) by RELAT_1:36
.= g * (id (rng f)) by A17, A14, TOPS_2:52
.= (id (rng g)) * g by A8, A17, TOPS_2:def 5
.= g by RELAT_1:54 ;
then A35: f . (((f ") * g) . t) = g . t by A29, A27, A32, FUNCT_1:13;
s2 in dom g by A12, A13, A29, BORSUK_1:43;
then ((f ") * g) . s2 = (f ") . q1 by A11, FUNCT_1:13
.= s1 by A2, A14, A31, A18, FUNCT_1:32 ;
then ((f ") * g) . s29 = s1 ;
then Ft < s1 by A25, A23, A33, A30, JORDAN5A:15, TOPMETR:20;
hence not g . t in Q by A7, A34, A35; :: thesis: verum