reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;
reserve I for non empty set,
  x,X,Y for ManySortedSet of I;
reserve I for set,
  x,X,Y,Z for ManySortedSet of I;

theorem Th132:
  X is non-empty & X [= Y implies X c= Y
proof
  assume that
A1: X is non-empty and
A2: X [= Y;
  deffunc F(object) = X.$1;
A3: for i being object st i in I holds F(i) <> {} by A1;
  consider f being ManySortedSet of I such that
A4: for i being object st i in I holds f.i in F(i)
from KuratowskiFunction(A3);
  let i be object such that
A5: i in I;
  let e be object;
  deffunc G(object) = IFEQ(i,$1,e,f.$1);
  consider g being Function such that
A6: dom g = I and
A7: for u being object st u in I holds g.u = G(u) from FUNCT_1:sch 3;
  reconsider g as ManySortedSet of I by A6,PARTFUN1:def 2,RELAT_1:def 18;
A8: g.i = IFEQ(i,i,e,f.i) by A5,A7
    .= e by FUNCOP_1:def 8;
  assume
A9: e in X.i;
  g in X
  proof
    let u be object such that
A10: u in I;
    per cases;
    suppose
      u = i;
      hence thesis by A8,A9;
    end;
    suppose
A11:  u <> i;
      g.u = IFEQ(i,u,e,f.u) by A7,A10
        .= f.u by A11,FUNCOP_1:def 8;
      hence thesis by A4,A10;
    end;
  end;
  then g in Y by A2;
  hence thesis by A5,A8;
end;
