reserve r,s,t,u for Real;

theorem Th33:
  for X being non empty RLTopStruct, a being Point of X, V being
  Subset of X holds transl(a,X).:V = a+V
proof
  let X be non empty RLTopStruct, a be Point of X, V be Subset of X;
  thus transl(a,X).:V c= a+V
  proof
    let y be object;
    assume y in transl(a,X).:V;
    then consider c being Point of X such that
A1: c in V and
A2: y = transl(a,X).c by FUNCT_2:65;
    y = a+c by A2,Def10;
    hence thesis by A1,Lm1;
  end;
  let x be object;
  assume x in a+V;
  then x in {a + u where u is Point of X: u in V} by RUSUB_4:def 8;
  then consider u being Point of X such that
A3: x = a+u and
A4: u in V;
  x = transl(a,X).u by A3,Def10;
  hence thesis by A4,FUNCT_2:35;
end;
