reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th68:
  f|Y is bounded iff ex p be Real st for c st c in Y /\ dom
  f holds |.(f/.c).|<= p
proof
A1: dom |.f.| = dom f by VALUED_1:def 11;
A2: dom(|.f.||Y) = Y /\ dom|.f.| by RELAT_1:61;
A3: |.f.||Y = |.f|Y.| by RFUNCT_1:46;
  hereby
    assume f|Y is bounded;
    then |.f.||Y is bounded by A3,Lm3;
    then consider p be Real such that
A4: for c being object st c in dom(|.f.||Y) holds |.(|.f.||Y).c.| <= p
    by RFUNCT_1:72;
    now
      let c such that
A5:   c in Y /\ dom f;
      c in dom f by A5,XBOOLE_0:def 4;
      then
A6:   f.c = f/.c by PARTFUN1:def 6;
      |.(|.f.||Y).c.| = |.|.f.|.c.| by A1,A2,A5,FUNCT_1:47
        .= |.|.(f.c).|.| by VALUED_1:18;
      hence |.(f/.c).| <= p by A1,A2,A4,A5,A6;
    end;
    hence
    ex p be Real st for c st c in Y /\ dom f holds |.(f/.c).| <= p;
  end;
  given p be Real such that
A7: for c st c in Y /\ dom f holds |.(f/.c).| <= p;
A8: dom |.f.| = dom f by VALUED_1:def 11;
  now
    let c be object such that
A9: c in dom(|.f.||Y);
A10: c in dom |.f.| by A9,RELAT_1:57;
    |.(|.f.||Y).c.| = |.|.f.|.c.| by A9,FUNCT_1:47
      .= |.|.(f.c).|.| by VALUED_1:18
      .= |.|.(f/.c).|.| by A1,A10,PARTFUN1:def 6;
    hence |.(|.f.||Y).c.| <= p by A2,A7,A8,A9;
  end;
  then |.f.||Y is bounded by RFUNCT_1:72;
  hence thesis by A3,Lm3;
end;
