reserve a for set;

theorem
  for L being lower-bounded sup-Semilattice holds
  MonSet L is full SubRelStr of (InclPoset Ids L)|^(the carrier of L)
proof
  let L be lower-bounded sup-Semilattice;
  set J = ((the carrier of L) --> InclPoset Ids L);
A1: the carrier of MonSet L c=
  the carrier of (InclPoset Ids L)|^(the carrier of L)
  proof
    let a be object;
    assume a in the carrier of MonSet L;
    then consider s be Function of L, InclPoset Ids L such that
A2: a = s and s is monotone
    and for x be Element of L holds s.x c= downarrow x by Def13;
    s in Funcs (the carrier of L, the carrier of InclPoset Ids L)
    by FUNCT_2:8;
    hence thesis by A2,YELLOW_1:28;
  end;
A3: the InternalRel of MonSet L c=
  the InternalRel of (InclPoset Ids L)|^(the carrier of L)
  proof
    let a,b be object;
    assume [a,b] in the InternalRel of MonSet L;
    then consider f,g be Function of L, InclPoset Ids L such that
A4: a = f and
A5: b = g and a in the carrier of MonSet L
    and b in the carrier of MonSet L and
A6: f <= g by Def13;
    set AG = product ((the carrier of L) --> InclPoset Ids L);
A7: AG = ((InclPoset Ids L) |^ the carrier of L) by YELLOW_1:def 5;
A8: f in Funcs (the carrier of L, the carrier of InclPoset Ids L) by FUNCT_2:8;
A9: g in Funcs (the carrier of L, the carrier of InclPoset Ids L) by FUNCT_2:8;
A10: f in the carrier of AG by A7,A8,YELLOW_1:28;
    reconsider f9 = f, g9 = g as Element of AG by A7,A8,A9,YELLOW_1:28;
A11: f9 in product Carrier ((the carrier of L) --> InclPoset Ids L)
    by A10,YELLOW_1:def 4;
    ex ff,gg being Function st ff = f9 & gg = g9 &
    for i be object st i in the carrier of L
    ex R being RelStr, xi,yi being Element of R
    st R = J.i & xi = ff.i & yi = gg.i & xi <= yi
    proof
      take f,g;
      thus f = f9 & g = g9;
      let i be object;
      assume
A12:  i in the carrier of L;
      then reconsider i9 = i as Element of L;
      take R = InclPoset Ids L;
      reconsider xi = f.i9, yi = g.i9 as Element of R;
      take xi, yi;
      thus R = J.i & xi = f.i & yi = g.i by A12,FUNCOP_1:7;
      reconsider i9 = i as Element of L by A12;
      ex a, b be Element of InclPoset Ids L st ( a = f.i9)&( b = g
      .i9)&( a <= b) by A6;
      hence thesis;
    end;
    then f9 <= g9 by A11,YELLOW_1:def 4;
    then [a,b] in the InternalRel of
    product ((the carrier of L) --> InclPoset Ids L) by A4,A5,ORDERS_2:def 5;
    hence thesis by YELLOW_1:def 5;
  end;
  set J = ((the carrier of L) --> InclPoset Ids L);
  the InternalRel of MonSet L =
  (the InternalRel of (InclPoset Ids L)|^(the carrier of L) )|_2
  the carrier of MonSet L
  proof
    let a,b be object;
    thus [a,b] in the InternalRel of MonSet L implies
    [a,b] in (the InternalRel of (InclPoset Ids L)|^(the carrier of L))|_2
    the carrier of MonSet L by A3,XBOOLE_0:def 4;
    assume
A13: [a,b] in (the InternalRel of (InclPoset Ids L)|^
    (the carrier of L) )|_2 the carrier of MonSet L;
    then
A14: [a,b] in (the InternalRel of (InclPoset Ids L)|^ (the carrier of L) )
    by XBOOLE_0:def 4;
A15: [a,b] in [:the carrier of MonSet L, the carrier of MonSet L:] by A13,
XBOOLE_0:def 4;
A16: a in the carrier of (InclPoset Ids L)|^(the carrier of L) by A14,
ZFMISC_1:87;
A17: b in the carrier of (InclPoset Ids L)|^(the carrier of L) by A14,
ZFMISC_1:87;
A18: a in the carrier of product J by A16,YELLOW_1:def 5;
    reconsider a9 = a, b9 = b as Element of product J by A16,A17,YELLOW_1:def 5
    ;
    [a9,b9] in (the InternalRel of product J) by A14,YELLOW_1:def 5;
    then
A19: a9 <= b9 by ORDERS_2:def 5;
    a9 in product Carrier J by A18,YELLOW_1:def 4;
    then consider f,g being Function such that
A20: f = a9 and
A21: g = b9 and
A22: for i be object st i in the carrier of L ex R being RelStr, xi,yi
being Element of R st R = ((the carrier of L) --> InclPoset Ids L).i & xi = f.i
    & yi = g.i & xi <= yi
    by A19,YELLOW_1:def 4;
A23: f in Funcs (the carrier of L, the carrier of InclPoset Ids L) by A16,A20,
YELLOW_1:28;
    g in Funcs (the carrier of L, the carrier of InclPoset Ids L) by A17,A21,
YELLOW_1:28;
    then reconsider f, g as Function of the carrier of L,
    the carrier of InclPoset Ids L by A23,FUNCT_2:66;
    reconsider f, g as Function of L, InclPoset Ids L;
    now
      take f, g;
      f <= g
      proof
        let j be set;
        assume j in the carrier of L;
        then reconsider j9 = j as Element of L;
        take f.j9, g.j9;
        ex R being RelStr, xi,yi being Element of R st ( R = ((the
carrier of L) --> InclPoset Ids L).j9)&( xi = f.j9)&( yi = g.j9)&( xi <= yi)
        by A22;
        hence thesis;
      end;
      hence a9 = f & b9 = g & a9 in the carrier of MonSet L &
      b9 in the carrier of MonSet L & f <= g by A15,A20,A21,ZFMISC_1:87;
    end;
    hence thesis by Def13;
  end;
  hence thesis by A1,A3,YELLOW_0:def 13,def 14;
end;
