let T be non empty interval SubSpace of R^1 ; :: thesis: for a, b being Point of T
for P, Q being Path of a,b holds P,Q are_homotopic
let a, b be Point of T; :: thesis: for P, Q being Path of a,b holds P,Q are_homotopic
let P, Q be Path of a,b; :: thesis: P,Q are_homotopic
take F = R1Homotopy P,Q; :: according to BORSUK_2:def 7 :: thesis: ( F is continuous & ( for b₁ being Element of the carrier of I[01] holds
( F . b₁,0 = P . b₁ & F . b₁,1 = Q . b₁ & F . 0 ,b₁ = a & F . 1,b₁ = b ) ) )
thus F is continuous by Lm8; :: thesis: for b₁ being Element of the carrier of I[01] holds
( F . b₁,0 = P . b₁ & F . b₁,1 = Q . b₁ & F . 0 ,b₁ = a & F . 1,b₁ = b )
thus for b₁ being Element of the carrier of I[01] holds
( F . b₁,0 = P . b₁ & F . b₁,1 = Q . b₁ & F . 0 ,b₁ = a & F . 1,b₁ = b ) by Lm9; :: thesis: verum