reserve r,s,t,u for Real;

theorem Th46:
  for X being non empty RLTopStruct, V being Subset of X, r being
  non zero Real holds mlt(r,X).:V = r*V
proof
  let X be non empty RLTopStruct, V be Subset of X, r be non zero Real;
  thus mlt(r,X).:V c= r*V
  proof
    let y be object;
    assume y in mlt(r,X).:V;
    then consider c being Point of X such that
A1: c in V and
A2: y = mlt(r,X).c by FUNCT_2:65;
    y = r*c by A2,Def13;
    hence thesis by A1;
  end;
  let x be object;
  assume x in r*V;
  then consider u being Point of X such that
A3: x = r*u and
A4: u in V;
  x = mlt(r,X).u by A3,Def13;
  hence thesis by A4,FUNCT_2:35;
end;
