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 Th3:
  for X being real-membered set, a being Real st a <> 0
  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;
  assume
A2: a <> 0;
A3: f is one-to-one
  proof
    let x,y be object;
    assume that
A4: x in dom f & y in dom f and
A5: f.x = f.y;
    reconsider x,y as Element of REAL by A1,A4,XREAL_0:def 1;
    f.x = a * x & f.y = a * y by A1,A4;
    hence thesis by A2,A5,XCMPLX_1:5;
  end;
  take f;
  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
A6:   x in dom f and
A7:   z = f.x by FUNCT_1:def 3;
      reconsider x9 = x as Element of REAL by A1,A6,XREAL_0:def 1;
      z = a * x9 by A1,A6,A7;
      hence thesis by A1,A6,MEMBER_1:193;
    end;
    let z be object;
    assume
A8: z in (a ** X);
    then reconsider z as Element of REAL;
    consider x being Complex such that
A9: z = a * x and
A10: x in X by A8,MEMBER_1:195;
    reconsider x as Element of REAL by A10,XREAL_0:def 1;
    f.x = z by A1,A10,A9;
    hence thesis by A1,A10,FUNCT_1:def 3;
  end;
  hence thesis by A1,A3;
end;
