let X be non empty TopSpace; :: thesis: for a, b, c being Point of X st a,b are_connected & c,a are_connected holds
for A1, A2 being Path of a,b
for B being Path of c,a st A1,A2 are_homotopic holds
A1,((- B) + B) + A2 are_homotopic
let a, b, c be Point of X; :: thesis: ( a,b are_connected & c,a are_connected implies for A1, A2 being Path of a,b
for B being Path of c,a st A1,A2 are_homotopic holds
A1,((- B) + B) + A2 are_homotopic )
assume that
A1: a,b are_connected and
A2: c,a are_connected ; :: thesis: for A1, A2 being Path of a,b
for B being Path of c,a st A1,A2 are_homotopic holds
A1,((- B) + B) + A2 are_homotopic
consider X being constant Path of a,a;
let A1, A2 be Path of a,b; :: thesis: for B being Path of c,a st A1,A2 are_homotopic holds
A1,((- B) + B) + A2 are_homotopic
let B be Path of c,a; :: thesis: ( A1,A2 are_homotopic implies A1,((- B) + B) + A2 are_homotopic )
A3: A1,X + A1 are_homotopic by A1, BORSUK_6:90;
assume A4: A1,A2 are_homotopic ; :: thesis: A1,((- B) + B) + A2 are_homotopic
(- B) + B,X are_homotopic by A2, BORSUK_6:94;
then ((- B) + B) + A2,X + A1 are_homotopic by A1, A4, BORSUK_6:83;
hence A1,((- B) + B) + A2 are_homotopic by A3, BORSUK_6:87; :: thesis: verum