reserve m,j,p,q,n,l for Element of NAT;
reserve e1,e2 for ExtReal;

theorem Th35:
  for F being non empty NAT-defined finite Function
  holds dom CutLastLoc F = (dom F) \ {LastLoc F}
proof
  let F be non empty NAT-defined finite Function;
A1: dom (LastLoc F .--> (F.LastLoc F)) = {LastLoc F};
  reconsider R = {[LastLoc F, F.LastLoc F]} as Relation;
A2: R = LastLoc F .--> (F.LastLoc F) by FUNCT_4:82;
  then
A3: dom R = {LastLoc F};
  thus dom CutLastLoc F c= (dom F) \ {LastLoc F}
  proof
    let x be object;
    assume x in dom CutLastLoc F;
    then consider y being object such that
A4: [x,y] in F \ R by A2,XTUPLE_0:def 12;
A5: not [x,y] in R by A4,XBOOLE_0:def 5;
A6: x in dom F by A4,XTUPLE_0:def 12;
    per cases by A5,TARSKI:def 1;
    suppose x <> LastLoc F;
      then not x in dom R by A3,TARSKI:def 1;
      hence thesis by A2,A6,XBOOLE_0:def 5;
    end;
    suppose
A7:   y <> F.LastLoc F;
      now
        assume x in dom R;
        then x = LastLoc F by A3,TARSKI:def 1;
        hence contradiction by A4,A7,FUNCT_1:1;
      end;
      hence thesis by A2,A6,XBOOLE_0:def 5;
    end;
  end;
  thus thesis by A1,XTUPLE_0:25;
end;
