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 Th70:
  f|Y is bounded_above iff ex r st for c being object st c in Y /\
  dom f holds f.c <= r
proof
  thus f|Y is bounded_above implies ex r st
   for c being object st c in Y /\ dom f
  holds f.c <= r
  proof
    given r being Real such that
A1: for p being object st p in dom(f|Y) holds (f|Y).p<r;
    take r;
    let c be object;
    assume c in Y /\ dom f;
    then
A2: c in dom(f|Y) by RELAT_1:61;
    then (f|Y).c < r by A1;
    hence thesis by A2,FUNCT_1:47;
  end;
  given r such that
A3: for c being object st c in Y /\ dom f holds f.c <= r;
  reconsider r1 = r+1 as Real;
  take r1;
  let p be object;
  assume
A4: p in dom(f|Y);
  then p in Y /\ dom f by RELAT_1:61;
  then f.p <= r by A3;
  then
A5: (f|Y).p <= r by A4,FUNCT_1:47;
  r < r1 by XREAL_1:29;
  hence thesis by A5,XXREAL_0:2;
end;
