reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;

theorem Th1:
  An = (1-lambda) * x1 + lambda * x2 & Bn = (1-mu) * x1 + mu * x2
  implies Bn - An = (mu-lambda)*(x2-x1)
  proof
    assume that
A1: An = (1-lambda) * x1 + lambda * x2 and
A2: Bn = (1-mu) * x1 + mu * x2;
A3: (1-lambda) * x1 = x1 - lambda * x1 by Lm1;
    Bn - An = (1-mu) * x1 + mu * x2 - (1-lambda)* x1 - lambda*x2
               by A1,A2,RVSUM_1:39
           .= x1 -mu * x1 + mu * x2 - (x1 -lambda * x1) - lambda * x2
               by A3,Lm1
           .= x1 -mu * x1 + mu * x2 - x1 + lambda * x1 - lambda * x2
               by RVSUM_1:41
           .= x1 - (mu * x1 - mu * x2) + -x1 + lambda * x1 - lambda * x2
               by RVSUM_1:41
           .= -(mu * x1 - mu * x2) + x1 - x1 + lambda * x1 - lambda * x2
           .= -(mu * x1 - mu * x2) + lambda * x1 - lambda * x2 by RVSUM_1:42
           .= mu * x2 - mu*x1 + lambda * x1 - lambda *x2 by RVSUM_1:35
           .= mu * x2 + (- mu * x1 + lambda * x1) +- lambda *x2 by RVSUM_1:15
           .= mu * x2 + ((-1)* mu * x1 + lambda * x1) +- lambda *x2
               by RVSUM_1:49
           .= mu * x2 + (- mu +lambda) * x1 +- lambda *x2 by RVSUM_1:50
           .= (- mu + lambda) * x1 + (mu*x2 + -lambda *x2) by RVSUM_1:15
           .= (-(mu-lambda))*x1 +(mu-lambda)*x2 by EUCLIDLP:11
           .= (mu-lambda)*(x2-x1) by EUCLIDLP:12;
    hence thesis;
  end;
