reserve x,y for object, X for set;

theorem Th1:
  for p being ManySortedSet of X st support p = {x} holds p = (X--> 0)+*(x,p.x)
proof
  let p be ManySortedSet of X;
  assume
A1: support p = {x};
A2: for y being object st y in dom p holds p.y=(X-->0)+*(x,p.x).y
  proof
    let y be object;
    assume y in dom p;
    then y in X;
    then
A3: y in dom (X-->0) by FUNCOP_1:13;
    per cases;
    suppose
      x=y;
      hence thesis by A3,FUNCT_7:31;
    end;
    suppose
A4:   x<>y;
      then not y in support p by A1,TARSKI:def 1;
      then p.y=0 by PRE_POLY:def 7;
      hence thesis by A4,Lm1;
    end;
  end;
  dom ((X-->0)+*(x,p.x)) = dom (X-->0) by FUNCT_7:30
    .= X by FUNCOP_1:13;
  then dom p = dom ((X-->0)+*(x,p.x)) by PARTFUN1:def 2;
  hence thesis by A2,FUNCT_1:2;
end;
