reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;

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;
