reserve i,j,k,n,m for Nat;

theorem Th3:
  for p,q being Point of TOP-REAL n, r being Real st 0 < r & p = (
  1-r)*p+r*q holds p = q
proof
  let p,q be Point of TOP-REAL n, r be Real such that
A1: 0 < r and
A2: p = (1-r)*p+r*q;
A3: (1-r)*p+r*p = ((1-r)+r)*p by RLVECT_1:def 6
    .= p by RLVECT_1:def 8;
  r*p = r*p + 0.TOP-REAL n by RLVECT_1:4
    .= r*p + ((1-r)*p + -(1-r)*p) by RLVECT_1:5
    .= r*q + (1-r)*p + -(1-r)*p by A2,A3,RLVECT_1:def 3
    .= r*q + ((1-r)*p + -(1-r)*p) by RLVECT_1:def 3
    .= r*q + 0.TOP-REAL n by RLVECT_1:5
    .= r*q by RLVECT_1:4;
  hence thesis by A1,RLVECT_1:36;
end;
