reserve r,s,t,u for Real;

theorem Th37:
  for X being LinearTopSpace, x being Point of X, V being Subset
  of X holds x+Int(V) = Int(x+V)
proof
  let X be LinearTopSpace, x be Point of X, V be Subset of X;
  x+Int(V) c= x+V by Th8,TOPS_1:16;
  hence x+Int(V) c= Int(x+V) by TOPS_1:24;
  let v be object;
  assume
A1: v in Int(x+V);
  then reconsider v as Point of X;
  consider Q being Subset of X such that
A2: Q is open and
A3: Q c= x+V and
A4: v in Q by A1,TOPS_1:22;
  -x+Q c= -x+(x+V) by A3,Th8;
  then -x+Q c= -x+x+V by Th6;
  then -x+Q c= 0.X+V by RLVECT_1:5;
  then -x+Q c= V by Th5;
  then
A5: x+Int(V) = {x + u where u is Point of X: u in Int V} & -x+Q c=Int(V) by A2,
RUSUB_4:def 8,TOPS_1:24;
  -x+v in -x+Q by A4,Lm1;
  then x+(-x+v) in x+Int(V) by A5;
  then x+-x+v in x+Int(V) by RLVECT_1:def 3;
  then 0.X+v in x+Int(V) by RLVECT_1:5;
  hence thesis;
end;
