 reserve a,b,c,x for Real;

theorem Andr1a:
  for C being non empty Subset of REAL holds
    rng (AffineMap (0,a) | C) = {a}
  proof
    let C be non empty Subset of REAL;
    set f = AffineMap (0,a) | C;
    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 = AffineMap (0,a).x by A1,FUNCT_1:49; then
      y = 0 * x + a by FCONT_1:def 4 .= a;
      hence thesis by TARSKI:def 1;
    end;
    let y be object;
    assume
a0: y in {a};
    reconsider o = the Element of C as Real;
    C c= REAL; then
    C c= dom AffineMap(0,a) by FUNCT_2:def 1; then
a1: dom f = C by RELAT_1:62;
    y = 0 * o + a by a0,TARSKI:def 1
     .= AffineMap (0,a).o by FCONT_1:def 4
     .= f.o by FUNCT_1:49;
    hence thesis by FUNCT_1:def 3,a1;
  end;
