
theorem Th18:
  for A being non empty Subset of REAL, x being Real, y being
  R_eal st x = y & 0 <= y holds sup(x ** A) = y * sup A
proof
  let A be non empty Subset of REAL, x being Real,
      y being R_eal such that
A1: x = y and
A2: 0 <= y;
  reconsider Y = x ** A as non empty Subset of REAL;
  per cases;
  suppose
A3: A is not bounded_above;
    per cases by A2;
    suppose
A4:   y = 0;
      then x ** A = {0} by A1,INTEGRA2:40;
      hence sup(x ** A) = 0 by XXREAL_2:11
        .= y * sup A by A4;
    end;
    suppose
A5:   y > 0;
      then Y is not bounded_above by A1,A3,Lm1;
      hence sup(x ** A) = +infty by XXREAL_2:73
        .= y * +infty by A5,XXREAL_3:def 5
        .= y * sup A by A3,XXREAL_2:73;
    end;
  end;
  suppose
    A is bounded_above;
    then reconsider X = A as non empty bounded_above real-membered set;
    reconsider u = upper_bound X as Real;
    x ** X is bounded_above by A1,A2,INTEGRA2:9;
    then reconsider Y as non empty bounded_above real-membered set;
    thus sup(x ** A) = upper_bound Y .= x * u by A1,A2,INTEGRA2:13
      .= y * sup A by A1,EXTREAL1:1;
  end;
end;
