let n be Element of NAT ; :: thesis: for T being non empty convex SubSpace of TOP-REAL n
for a, b being Point of T
for P, Q being Path of a,b holds ConvexHomotopy (P,Q) is Homotopy of P,Q
let T be non empty convex SubSpace of TOP-REAL n; :: thesis: for a, b being Point of T
for P, Q being Path of a,b holds ConvexHomotopy (P,Q) is Homotopy of P,Q
let a, b be Point of T; :: thesis: for P, Q being Path of a,b holds ConvexHomotopy (P,Q) is Homotopy of P,Q
let P, Q be Path of a,b; :: thesis: ConvexHomotopy (P,Q) is Homotopy of P,Q
thus P,Q are_homotopic by Th2; :: according to BORSUK_6:def 11 :: thesis: ( ConvexHomotopy (P,Q) is continuous & ( for b₁ being Element of the carrier of I[01] holds
( (ConvexHomotopy (P,Q)) . (b₁,0) = P . b₁ & (ConvexHomotopy (P,Q)) . (b₁,1) = Q . b₁ & (ConvexHomotopy (P,Q)) . (0,b₁) = a & (ConvexHomotopy (P,Q)) . (1,b₁) = b ) ) )
thus ConvexHomotopy (P,Q) is continuous ; :: thesis: for b₁ being Element of the carrier of I[01] holds
( (ConvexHomotopy (P,Q)) . (b₁,0) = P . b₁ & (ConvexHomotopy (P,Q)) . (b₁,1) = Q . b₁ & (ConvexHomotopy (P,Q)) . (0,b₁) = a & (ConvexHomotopy (P,Q)) . (1,b₁) = b )
thus for b₁ being Element of the carrier of I[01] holds
( (ConvexHomotopy (P,Q)) . (b₁,0) = P . b₁ & (ConvexHomotopy (P,Q)) . (b₁,1) = Q . b₁ & (ConvexHomotopy (P,Q)) . (0,b₁) = a & (ConvexHomotopy (P,Q)) . (1,b₁) = b ) by Lm6; :: thesis: verum