theorem Th25:
  W.(a,p) = x & W.(a,b) = v implies W.(a,(p+*(m,b))) = (x+*(m,v))
proof
  assume that
A1: W.(a,p) = x and
A2: W.(a,b) = v;
  set q = (p+*(m,b));
  set y = W.(a,q), z = (x+*(m,v));
  for k being Nat of n holds y.k = z.k
  proof
    let k be Nat of n;
    now
      per cases;
      suppose
A3:     k = m;
        thus y.k = W.(a,q.k) by Def9
          .= W.(a,b) by A3,Th10
          .= z.k by A2,A3,Th13;
      end;
      suppose
A4:     k <> m;
        thus y.k = W.(a,q.k) by Def9
          .= W.(a,p.k) by A4,FUNCT_7:32
          .= x.k by A1,Def9
          .= z.k by A4,FUNCT_7:32;
      end;
    end;
    hence thesis;
  end;
  hence thesis by Th14;
end;
