reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;
reserve f for real-valued Function;

theorem Th72:
  f is bounded iff ex r st for c being object st c in dom f holds |.f.c.|<=r
proof
  thus f is bounded implies ex r st for c being object st c in dom f
    holds |.f.c.|<=r
  proof
    assume
A1: f is bounded;
    then consider r1 such that
A2: for c being object st c in dom f holds f.c < r1 by SEQ_2:def 1;
    consider r2 such that
A3: for c being object st c in dom f holds r2 < f.c by A1,SEQ_2:def 2;
    take g=|.r1.|+|.r2.|;
    let c be object such that
A4: c in dom f;
A5: 0 <= |.r2.| by COMPLEX1:46;
    r1 <= |.r1.| by ABSVALUE:4;
    then f.c <= |.r1.| by A2,A4,XXREAL_0:2;
    then
A6: f.c + (0 qua Real)<= |.r1.|+|.r2.| by A5,XREAL_1:7;
A7: 0 <= |.r1.| by COMPLEX1:46;
    -|.r2.| <= r2 by ABSVALUE:4;
    then -|.r2.| <= f.c by A3,A4,XXREAL_0:2;
    then
A8: -|.r1.|+-|.r2.| <= ((0 qua Real) + f.c)
by A7,XREAL_1:7;
    -|.r1.|+-|.r2.| = -g;
    hence thesis by A6,A8,ABSVALUE:5;
  end;
  given r such that
A9: for c being object st c in dom f holds |.f.c.| <= r;
  thus f is bounded_above
  proof
    take r+1;
    let c be object;
    assume c in dom f;
    then
A10: |.f.c.| < r+1 by A9,XREAL_1:39;
    f.c <= |.f.c.| by ABSVALUE:4;
    hence thesis by A10,XXREAL_0:2;
  end;
  take -(r+1);
  let c be object;
  assume c in dom f;
  then |.f.c.| < r+1 by A9,XREAL_1:39;
  then
A11: -(r+1) < -|.f.c.| by XREAL_1:24;
  -|.f.c.| <= f.c by ABSVALUE:4;
  hence thesis by A11,XXREAL_0:2;
end;
