reserve r,s,t,u for Real;

theorem Th51:
  for X being LinearTopSpace, V being Subset of X, r be non zero
  Real holds r*Int(V) = Int(r*V)
proof
  let X be LinearTopSpace, V be Subset of X, r be non zero Real;
  r*Int(V) c= r*V by CONVEX1:39,TOPS_1:16;
  hence r*Int(V) c= Int(r*V) by Th49,TOPS_1:24;
  let x be object;
  assume
A1: x in Int(r*V);
  then reconsider x as Point of X;
  consider Q being Subset of X such that
A2: Q is open and
A3: Q c= r*V and
A4: x in Q by A1,TOPS_1:22;
  r"*Q c= r"*(r*V) by A3,CONVEX1:39;
  then r"*Q c= r"*r*V by CONVEX1:37;
  then r"*Q c= 1*V by XCMPLX_0:def 7;
  then
A5: r"*Q c= V by CONVEX1:32;
  r"*x in r"*Q & r"*Q is open by A2,A4,Th49;
  then r"*x in Int(V) by A5,TOPS_1:22;
  then r*(r"*x) in r*Int(V);
  then r*r"*x in r*Int(V) by RLVECT_1:def 7;
  then 1*x in r*Int(V) by XCMPLX_0:def 7;
  hence thesis by RLVECT_1:def 8;
end;
