reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;

theorem Th1:
  for X being real-membered set, a being Real holds X, a ++ X are_equipotent
proof
  let X be real-membered set, a be Real;
  deffunc F(Real) = a + $1;
  consider f being Function such that
A1: dom f = X & for x being Element of REAL st x in X holds f.x = F(x)
  from CLASSES1:sch 2;
A2: rng f = a ++ X
  proof
    thus rng f c= a ++ X
    proof
      let z be object;
      assume z in rng f;
      then consider x being object such that
A3:   x in dom f and
A4:   z = f.x by FUNCT_1:def 3;
      reconsider x as Real by A1,A3;
      reconsider x as Element of REAL by XREAL_0:def 1;
      a + x  in REAL by XREAL_0:def 1;
      then reconsider z9= z as Element of REAL by A1,A3,A4;
      z9 = a + x by A1,A3,A4;
      hence thesis by A1,A3,MEMBER_1:141;
    end;
    let z be object;
    assume
A5: z in (a ++ X);
    then reconsider z as Element of REAL;
    consider x being Complex such that
A6: z = a + x and
A7: x in X by A5,MEMBER_1:143;
    X c= REAL by MEMBERED:3; then
    reconsider x as Element of REAL by A7;
    f.x = z by A1,A7,A6;
    hence thesis by A1,A7,FUNCT_1:def 3;
  end;
  take f;
  f is one-to-one
  proof
    let x,y be object;
    assume that
A8: x in dom f and
A9: y in dom f and
A10: f.x = f.y;
    reconsider x,y as Element of REAL by A1,A8,A9,XREAL_0:def 1;
    f.x = a + x by A1,A8;
    then a + x = a + y by A1,A9,A10;
    hence thesis;
  end;
  hence thesis by A1,A2;
end;
