
theorem Th11:
for f be PartFunc of REAL,COMPLEX holds
    f is bounded iff (Re f is bounded & Im f is bounded)
proof
let f be PartFunc of REAL,COMPLEX;
thus f is bounded implies (Re f is bounded & Im f is bounded)
  proof
    assume f is bounded; then
    consider r being Real such that
  A1: for y be set st y in dom f holds |.f.y.| < r by COMSEQ_2:def 3;
    now
      let y be set;
      assume A2:y in dom (Re f); then
    A3:y in dom f by COMSEQ_3:def 3;
      |.Re (f.y).| <= |.f.y.| by COMPLEX1:79; then
      |.Re (f.y).| < r by A1,A3,XXREAL_0:2;
      hence |.(Re f).y.| < r by A2,COMSEQ_3:def 3;
    end;
    hence Re f is bounded by COMSEQ_2:def 3;
    now
      let y be set;
      assume A4:y in dom (Im f); then
    A5:y in dom f by COMSEQ_3:def 4;
      |.Im (f.y).| <= |.f.y.| by COMPLEX1:79; then
      |.Im (f.y).| < r by A1,A5,XXREAL_0:2;
      hence |.(Im f).y.| < r by A4,COMSEQ_3:def 4;
    end;
    hence Im f is bounded by COMSEQ_2:def 3;
  end;
thus (Re f is bounded & Im f is bounded) implies f is bounded
  proof
    assume A6: Re f is bounded & Im f is bounded; then
    consider r1 being Real such that
  A7: for y be set st y in dom (Re f) holds |.(Re f).y.| < r1 by
COMSEQ_2:def 3;
    consider r2 being Real such that
  A8: for y be set st y in dom (Im f)
         holds |.(Im f).y.| < r2 by A6,COMSEQ_2:def 3;
    now let y be set;
      assume A9:y in dom f; then
    A10: y in dom (Re f) by COMSEQ_3:def 3; then
      |.(Re f).y.| < r1 by A7; then
    A11: |.Re (f.y).| < r1 by A10,COMSEQ_3:def 3;
    A12: y in dom (Im f) by A9,COMSEQ_3:def 4; then
      |.(Im f).y.| < r2 by A8; then
    A13: |.Im (f.y).| < r2 by A12,COMSEQ_3:def 4;
    A14: |.f.y.| <= |.Re (f.y).| + |.Im(f.y).| by COMPLEX1:78;
      |.Re (f.y).| +  |.Im(f.y).| < r1+r2 by A11,A13,XREAL_1:8;
      hence |.f.y.| < r1+r2 by A14,XXREAL_0:2;
    end;
    hence f is bounded by COMSEQ_2:def 3;
  end;
end;
