reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;

theorem Th88:
  for f,g being PartFunc of X,Y st dom g c= dom f & TotFuncs f c=
  TotFuncs g holds g c= f
proof
  let f,g be PartFunc of X,Y such that
A1: dom g c= dom f;
  now
    per cases;
    suppose Y = {} & X <> {};
      hence thesis;
    end;
    suppose
A2:   Y = {} implies X = {};
      thus TotFuncs f c= TotFuncs g implies g c= f
      proof
        assume
A3:     TotFuncs f c= TotFuncs g;
        for x being object st x in dom g holds g.x = f.x
        proof
          let x be object;
          consider h being Function of X,Y such that
A4:       f tolerates h by A2,Th76;
          h in TotFuncs f by A2,A4,PARTFUN1:def 5;
          then
A5:       g tolerates h by A3,PARTFUN1:71;
          assume x in dom g;
          then x in dom f /\ dom g by A1,XBOOLE_0:def 4;
          hence thesis by A5,A2,A4,PARTFUN1:67,def 4;
        end;
        hence thesis by A1,GRFUNC_1:2;
      end;
    end;
  end;
  hence thesis;
end;
