let T be non empty pathwise_connected TopSpace; for a1, b1, c1, d1 being Point of T
for A being Path of a1,b1
for B being Path of c1,d1
for C being Path of a1,c1 holds (((A + (- A)) + C) + B) + (- B),C are_homotopic
let a1, b1, c1, d1 be Point of T; for A being Path of a1,b1
for B being Path of c1,d1
for C being Path of a1,c1 holds (((A + (- A)) + C) + B) + (- B),C are_homotopic
A1:
c1,d1 are_connected
by BORSUK_2:def 3;
( a1,b1 are_connected & a1,c1 are_connected )
by BORSUK_2:def 3;
hence
for A being Path of a1,b1
for B being Path of c1,d1
for C being Path of a1,c1 holds (((A + (- A)) + C) + B) + (- B),C are_homotopic
by A1, Th39; verum