let X be non empty TopSpace; for a, b being Point of X st a,b are_connected holds
for P, Q being Path of a,b holds
( Q in Class ((EqRel (X,a,b)),P) iff P,Q are_homotopic )
let a, b be Point of X; ( a,b are_connected implies for P, Q being Path of a,b holds
( Q in Class ((EqRel (X,a,b)),P) iff P,Q are_homotopic ) )
set E = EqRel (X,a,b);
assume A1:
a,b are_connected
; for P, Q being Path of a,b holds
( Q in Class ((EqRel (X,a,b)),P) iff P,Q are_homotopic )
let P, Q be Path of a,b; ( Q in Class ((EqRel (X,a,b)),P) iff P,Q are_homotopic )
A2:
( not EqRel (X,a,b) is empty & EqRel (X,a,b) is total & EqRel (X,a,b) is symmetric & EqRel (X,a,b) is transitive )
by A1, Lm3;
hereby ( P,Q are_homotopic implies Q in Class ((EqRel (X,a,b)),P) )
assume
Q in Class (
(EqRel (X,a,b)),
P)
;
P,Q are_homotopic then
[Q,P] in EqRel (
X,
a,
b)
by A2, EQREL_1:19;
hence
P,
Q are_homotopic
by A1, Def3;
verum
end;
assume
P,Q are_homotopic
; Q in Class ((EqRel (X,a,b)),P)
then
[Q,P] in EqRel (X,a,b)
by A1, Def3;
hence
Q in Class ((EqRel (X,a,b)),P)
by A2, EQREL_1:19; verum