theorem Th14:
  w = v implies a * w = a * v
proof
  assume
A1: w = v;
  reconsider aa=a as Element of REAL by XREAL_0:def 1;
  aa * w = ((the Mult of V) | [:REAL, the carrier of W:]).[aa,w] by Def2;
   then aa * w = aa * v by A1,FUNCT_1:49;
  hence thesis;
end;
