let X be non empty TopSpace; :: thesis: for a, b, c being Point of X st a,b are_connected & b,c are_connected holds
for A1, A2 being Path of a,b
for B being Path of b,c st A1,A2 are_homotopic holds
A1,(A2 + B) + (- B) are_homotopic
let a, b, c be Point of X; :: thesis: ( a,b are_connected & b,c are_connected implies for A1, A2 being Path of a,b
for B being Path of b,c st A1,A2 are_homotopic holds
A1,(A2 + B) + (- B) are_homotopic )
assume that
A1: a,b are_connected and
A2: b,c are_connected ; :: thesis: for A1, A2 being Path of a,b
for B being Path of b,c st A1,A2 are_homotopic holds
A1,(A2 + B) + (- B) are_homotopic
set X = the constant Path of b,b;
let A1, A2 be Path of a,b; :: thesis: for B being Path of b,c st A1,A2 are_homotopic holds
A1,(A2 + B) + (- B) are_homotopic
let B be Path of b,c; :: thesis: ( A1,A2 are_homotopic implies A1,(A2 + B) + (- B) are_homotopic )
A3: A1,A1 + the constant Path of b,b are_homotopic by A1, BORSUK_6:80;
assume A4: A1,A2 are_homotopic ; :: thesis: A1,(A2 + B) + (- B) are_homotopic
B + (- B), the constant Path of b,b are_homotopic by A2, BORSUK_6:84;
then A2 + (B + (- B)),A1 + the constant Path of b,b are_homotopic by A1, A4, BORSUK_6:75;
then A5: A2 + (B + (- B)),A1 are_homotopic by A3, BORSUK_6:79;
A2 + (B + (- B)),(A2 + B) + (- B) are_homotopic by A1, A2, BORSUK_6:73;
hence A1,(A2 + B) + (- B) are_homotopic by A5, BORSUK_6:79; :: thesis: verum