reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;

theorem Th72:
  for f being PartFunc of X,Y holds f is total iff TotFuncs f = { f}
proof
  let f be PartFunc of X,Y;
  thus f is total implies TotFuncs f = {f}
  proof
    assume
A1: f is total;
    for g being object holds g in TotFuncs f iff f = g
    proof
      let g be object;
      thus g in TotFuncs f implies f = g
      proof
        assume g in TotFuncs f;
        then
        ex g9 being PartFunc of X,Y st g9 = g & g9 is total & f tolerates
        g9 by Def5;
        hence thesis by A1,Th66;
      end;
      thus thesis by A1,Def5;
    end;
    hence thesis by TARSKI:def 1;
  end;
  assume TotFuncs f = {f};
  then f in TotFuncs f by TARSKI:def 1;
  hence thesis by Th70;
end;
