
theorem Th63:
  for x be Element of COMPLEX holds COMPLEX --> x is_continuous_on COMPLEX
proof
  let x be Element of COMPLEX;
A1: now
    let x1 be Complex;
    let r be Real;
    assume that
A2:  x1 in COMPLEX and
A3: 0 < r;
    take s=r;
    thus 0 < s by A3;
    let x2 be Complex;
    assume that
A4:  x2 in COMPLEX and
    |.x2-x1.| < s;
    reconsider xx1=x1, xx2=x2 as Element of COMPLEX by A2,A4;
    (COMPLEX --> x)/.xx1 = x & (COMPLEX --> x)/.xx2 = x by FUNCOP_1:7;
    hence |.(COMPLEX --> x)/.x2 - (COMPLEX --> x)/.x1 .| < r by A3,COMPLEX1:44;
  end;
  dom (COMPLEX --> x) = COMPLEX by FUNCOP_1:13;
  hence thesis by A1,CFCONT_1:39;
end;
