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
  (Y c= X & f|X is bounded_above implies f|Y is bounded_above) & (Y c= X
& f|X is bounded_below implies f|Y is bounded_below) & (Y c= X & f|X is bounded
  implies f|Y is bounded)
proof
  thus
A1: Y c= X & f|X is bounded_above implies f|Y is bounded_above
  proof
    assume that
A2: Y c= X and
A3: f|X is bounded_above;
    consider r such that
A4: for c being object st c in X /\ dom f holds f.c <= r by A3,Th70;
    now
      take r;
      let c be object;
      assume
A5:   c in Y /\ dom f;
      then
A6:   c in dom f by XBOOLE_0:def 4;
      c in Y by A5,XBOOLE_0:def 4;
      then c in X /\ dom f by A2,A6,XBOOLE_0:def 4;
      hence f.c <= r by A4;
    end;
    hence thesis by Th70;
  end;
  thus
A7: Y c= X & f|X is bounded_below implies f|Y is bounded_below
  proof
    assume that
A8: Y c= X and
A9: f|X is bounded_below;
    consider r such that
A10: for c being object st c in X /\ dom f holds r <= f.c by A9,Th71;
    now
      take r;
      let c be object;
      assume
A11:  c in Y /\ dom f;
      then
A12:  c in dom f by XBOOLE_0:def 4;
      c in Y by A11,XBOOLE_0:def 4;
      then c in X /\ dom f by A8,A12,XBOOLE_0:def 4;
      hence r <= f.c by A10;
    end;
    hence thesis by Th71;
  end;
  assume that
A13: Y c= X and
A14: f|X is bounded;
  thus thesis by A1,A7,A13,A14;
end;
