 reserve a,b,c,x for Real;

theorem
  rng AffineMap (0,a) = {a}
  proof
    set f = AffineMap (0,a);
    thus rng f c= {a}
    proof
      let y be object;
      assume y in rng f; then
      consider x being object such that
A1:   x in dom f & y = f.x by FUNCT_1:def 3;
      reconsider x as Real by A1;
      y = 0 * x + a by A1,FCONT_1:def 4 .= a;
      hence thesis by TARSKI:def 1;
    end;
    let y be object;
    assume
a0: y in {a};
    0 in REAL by XREAL_0:def 1; then
A1: 0 in dom f by FUNCT_2:def 1;
    y = 0 * 0 + a by a0,TARSKI:def 1
     .= f.0 by FCONT_1:def 4;
    hence thesis by FUNCT_1:def 3,A1;
  end;
