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 R1Homotopy P,Q is Homotopy of P,Q
let a, b be Point of T; :: thesis: for P, Q being Path of a,b holds R1Homotopy P,Q is Homotopy of P,Q
let P, Q be Path of a,b; :: thesis: R1Homotopy P,Q is Homotopy of P,Q
thus P,Q are_homotopic by Th16; :: according to BORSUK_6:def 13 :: thesis: ( R1Homotopy P,Q is continuous & ( for b₁ being Element of the carrier of I[01] holds
( (R1Homotopy P,Q) . b₁,0 = P . b₁ & (R1Homotopy P,Q) . b₁,1 = Q . b₁ & (R1Homotopy P,Q) . 0 ,b₁ = a & (R1Homotopy P,Q) . 1,b₁ = b ) ) )
thus R1Homotopy P,Q is continuous ; :: thesis: for b₁ being Element of the carrier of I[01] holds
( (R1Homotopy P,Q) . b₁,0 = P . b₁ & (R1Homotopy P,Q) . b₁,1 = Q . b₁ & (R1Homotopy P,Q) . 0 ,b₁ = a & (R1Homotopy P,Q) . 1,b₁ = b )
thus for b₁ being Element of the carrier of I[01] holds
( (R1Homotopy P,Q) . b₁,0 = P . b₁ & (R1Homotopy P,Q) . b₁,1 = Q . b₁ & (R1Homotopy P,Q) . 0 ,b₁ = a & (R1Homotopy P,Q) . 1,b₁ = b ) by Lm9; :: thesis: verum