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

theorem Th37:
  for F being non empty NAT-defined finite Function
  holds card CutLastLoc F = card F - 1
proof
  let F be non empty NAT-defined finite Function;
  LastLoc F .--> F.LastLoc F c= F
  proof
    let a, b be object;
    assume [a,b] in LastLoc F .--> F.LastLoc F;
    then [a,b] in {[LastLoc F,F.LastLoc F]} by FUNCT_4:82;
    then
A1: [a,b] = [LastLoc F,F.LastLoc F] by TARSKI:def 1;
    LastLoc F in dom F by XXREAL_2:def 8;
    hence thesis by A1,FUNCT_1:def 2;
  end;
  hence card CutLastLoc F = card F - card (LastLoc F .--> F.LastLoc F)
  by CARD_2:44
    .= card F - card {[LastLoc F,F.LastLoc F]} by FUNCT_4:82
    .= card F - 1 by CARD_1:30;
end;
