
theorem Th07:
  for a,b being Real
  for P,Q being Element of TOP-REAL 2 st P <> Q &
  (1 - a) * P + a * Q = (1 - b) * P + b * Q holds
  a = b
  proof
    let a,b be Real;
    let P,Q be Element of TOP-REAL 2;
    assume that
A1: P <> Q and
A2: (1 - a) * P + a * Q = (1 - b) * P + b * Q;
    reconsider PR2 = P, QR2= Q as Element of REAL 2 by EUCLID:22;
    reconsider R2 = (1-a)*PR2+a*QR2 as Element of TOP-REAL 2 by EUCLID:22;
    per cases by A1,Th06;
    suppose P.1 <> Q.1;
      then
A3:   QR2.1 - PR2.1 <> 0;
      0 = R2.1 - R2.1
       .= (R2 - R2).1 by RVSUM_1:27
       .= ((b - a) * (QR2 - PR2)).1 by A2,EUCLID12:1
       .= (b - a) * (QR2 - PR2).1 by RVSUM_1:44
       .= (b - a) * (QR2.1 - PR2.1) by RVSUM_1:27;
      then (b - a) = 0 by A3;
      hence thesis;
    end;
    suppose P.2 <> Q.2; then
A4:   QR2.2 - PR2.2 <> 0;
      0 = R2.2 - R2.2
       .= (R2 - R2).2 by RVSUM_1:27
       .= ((b - a) * (QR2 - PR2)).2 by A2,EUCLID12:1
       .= (b - a) * (QR2 - PR2).2 by RVSUM_1:44
       .= (b - a) * (QR2.2 - PR2.2) by RVSUM_1:27;
      then (b - a) = 0 by A4;
      hence thesis;
    end;
  end;
