let X be non empty TopSpace; 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; ( 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
; 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
set X = the constant Path of a,a;
let A1, A2 be Path of a,b; 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; ( A1,A2 are_homotopic implies A1,((- B) + B) + A2 are_homotopic )
A3:
A1, the constant Path of a,a + A1 are_homotopic
by A1, BORSUK_6:82;
assume A4:
A1,A2 are_homotopic
; A1,((- B) + B) + A2 are_homotopic
(- B) + B, the constant Path of a,a are_homotopic
by A2, BORSUK_6:86;
then
((- B) + B) + A2, the constant Path of a,a + A1 are_homotopic
by A1, A4, BORSUK_6:75;
hence
A1,((- B) + B) + A2 are_homotopic
by A3, BORSUK_6:79; verum