let T be non empty pathwise_connected TopSpace; :: thesis: for a1, b1, c1, d1, e1, f1 being Point of T
for A being Path of a1,b1
for B being Path of b1,c1
for C being Path of c1,d1
for D being Path of d1,e1
for E being Path of f1,c1 holds (A + (B + C)) + D,((A + B) + (- E)) + ((E + C) + D) are_homotopic
let a1, b1, c1, d1, e1, f1 be Point of T; :: thesis: for A being Path of a1,b1
for B being Path of b1,c1
for C being Path of c1,d1
for D being Path of d1,e1
for E being Path of f1,c1 holds (A + (B + C)) + D,((A + B) + (- E)) + ((E + C) + D) are_homotopic
A1: ( c1,d1 are_connected & d1,e1 are_connected ) by BORSUK_2:def 3;
A2: a1,f1 are_connected by BORSUK_2:def 3;
( a1,b1 are_connected & b1,c1 are_connected ) by BORSUK_2:def 3;
hence for A being Path of a1,b1
for B being Path of b1,c1
for C being Path of c1,d1
for D being Path of d1,e1
for E being Path of f1,c1 holds (A + (B + C)) + D,((A + B) + (- E)) + ((E + C) + D) are_homotopic by A1, A2, Th43; :: thesis: verum