reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th27:
  p1<>0.TOP-REAL n & p2<>0.TOP-REAL n implies
  ex R being Function of TOP-REAL n, TOP-REAL n st R is being_homeomorphism &
  R .: Plane(p1,0.TOP-REAL n) = Plane(p2,0.TOP-REAL n)
proof
  assume p1<>0.TOP-REAL n; then
  consider B1 be linearly-independent Subset of TOP-REAL n such that
A1: card B1 = n - 1 & [#]Lin(B1) = Plane(p1,0.TOP-REAL n) by Th26;
  assume p2<>0.TOP-REAL n; then
  consider B2 be linearly-independent Subset of TOP-REAL n such that
A2: card B2 = n - 1 & [#]Lin(B2) = Plane(p2,0.TOP-REAL n) by Th26;
  consider M be Matrix of n,F_Real such that
A3: M is invertible & (Mx2Tran M).:[#]Lin B1 = [#]Lin B2 by A1,A2,MATRTOP2:22;
  reconsider M as invertible Matrix of n,F_Real by A3;
  take Mx2Tran M;
  thus thesis by A1,A2,A3;
end;
