let T be non empty arcwise_connected TopSpace; :: thesis: for a1, b1, c1 being Point of T
for A1, A2 being Path of a1,b1
for B being Path of b1,c1 st A1,A2 are_homotopic holds
A1,(A2 + B) + (- B) are_homotopic
let a1, b1, c1 be Point of T; :: thesis: for A1, A2 being Path of a1,b1
for B being Path of b1,c1 st A1,A2 are_homotopic holds
A1,(A2 + B) + (- B) are_homotopic
let A1, A2 be Path of a1,b1; :: thesis: for B being Path of b1,c1 st A1,A2 are_homotopic holds
A1,(A2 + B) + (- B) are_homotopic
( a1,b1 are_connected & b1,c1 are_connected ) by BORSUK_2:def 3;
hence for B being Path of b1,c1 st A1,A2 are_homotopic holds
A1,(A2 + B) + (- B) are_homotopic by Th20; :: thesis: verum