reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;

theorem
  for n being Nat
  for u,v being Element of TOP-REAL n
  for a,b being Real st
  (1 - a) * u + (a * v) = (1 - b) * v + (b * u)
  holds (1 - (a + b)) * u = (1 - (a + b)) * v
  proof
    let n be Nat;
    let u,v be Element of TOP-REAL n;
    let a,b be Real;
    assume
A1: (1 - a) * u + (a * v) = (1 - b) * v + (b * u);
    reconsider ru = u ,rv = v as Element of REAL n by EUCLID:22;
A2: (1 - a) * ru + a * rv - a * rv = (1 - a) * ru
    proof
      (1 - a) * ru in REAL n & a * rv in REAL n;
      then reconsider t1 = (1 - a) * u,
                      t2 = a * v as Element of n-tuples_on REAL
                        by EUCLID:def 1;
      (1 - a) * ru + a * rv - a * rv = t1 + t2 - t2
                                    .= t1 by RVSUM_1:42;
      hence thesis;
    end;
A3: (1 - b) * rv - a * rv + b * ru - b * ru = (1 - b) * rv - a * rv
    proof
      reconsider t1 = (1 - b) * rv - a * rv,
                 t2 = b * ru as Element of n-tuples_on REAL by EUCLID:def 1;
      (1 - b) * rv - a * rv + b * ru - b * ru = t1 + t2 - t2
                                             .= t1 by RVSUM_1:42;
      hence thesis;
    end;
    (1 - a) * ru = (1 - b) * rv + b * ru + -( a * rv) by A1,A2,RVSUM_1:31
                .= ((1 - b) * rv + - (a * rv)) + (b * ru) by RVSUM_1:15;
    then (1 - a) * ru - b * ru = (1 - b) * rv - a * rv by A3,RVSUM_1:31;
    then (1 - a) * ru + - (b * ru) = (1 - b) * rv - a * rv by RVSUM_1:31
                                  .= (1 - b) * rv + -(a * rv) by RVSUM_1:31;
    then (1 - a) * ru + (-1) * (b * ru) = (1 - b) * rv + -(a * rv)
      by RVSUM_1:54;
    then (1 - a) * ru + ((-1) * b) * ru = (1 - b) * rv + -(a * rv)
      by RVSUM_1:49;
    then ((1 - a) + (-1) * b) * ru = (1 - b) * rv + -(a * rv) by RVSUM_1:50;
    then (1 - a - b) * ru = (1 - b) * rv + ((-1) *(a * rv)) by RVSUM_1:54
                         .= (1 - b) * rv + ((-1) * a) * rv by RVSUM_1:49
                         .= ((1 - b) + (-1) * a) * rv by RVSUM_1:50;
    hence thesis;
  end;
