reserve T for non empty Abelian
  add-associative right_zeroed right_complementable RLSStruct,
  X,Y,Z,B,C,B1,B2 for Subset of T,
  x,y,p for Point of T;
reserve t,s,r1 for Real;
reserve n for Element of NAT;
reserve X,Y,B1,B2 for Subset of TOP-REAL n;
reserve x,y for Point of TOP-REAL n;

theorem Th60:
  (t*s)(.)X = t(.)(s(.)X)
proof
  thus (t*s)(.)X c= t(.)(s(.)X)
  proof
    let x be object;
    assume x in (t*s)(.)X;
    then consider z being Point of TOP-REAL n such that
A1: x = (t*s) * z & z in X;
    x = t*(s*z) & s*z in s(.)X by A1,RLVECT_1:def 7;
    hence thesis;
  end;
  let x be object;
  assume x in t(.)(s(.)X);
  then consider z being Point of TOP-REAL n such that
A2: x = t * z and
A3: z in s(.)X;
  consider z1 being Point of TOP-REAL n such that
A4: z = s * z1 and
A5: z1 in X by A3;
  x = (t*s)*z1 by A2,A4,RLVECT_1:def 7;
  hence thesis by A5;
end;
