let L be Lattice; :: thesis: ( dom () = the carrier of L & rng () = the carrier of L & field () = the carrier of L )
now :: thesis: for x being object holds
( ( x in the carrier of L implies ex y being object st [x,y] in LattRel L ) & ( ex y being object st [x,y] in LattRel L implies x in the carrier of L ) )
let x be object ; :: thesis: ( ( x in the carrier of L implies ex y being object st [x,y] in LattRel L ) & ( ex y being object st [x,y] in LattRel L implies x in the carrier of L ) )
thus ( x in the carrier of L implies ex y being object st [x,y] in LattRel L ) :: thesis: ( ex y being object st [x,y] in LattRel L implies x in the carrier of L )
proof
assume x in the carrier of L ; :: thesis: ex y being object st [x,y] in LattRel L
then reconsider p = x as Element of L ;
[p,p] in LattRel L ;
hence ex y being object st [x,y] in LattRel L ; :: thesis: verum
end;
given y being object such that A1: [x,y] in LattRel L ; :: thesis: x in the carrier of L
consider p, q being Element of L such that
A2: [x,y] = [p,q] and
p [= q by A1;
x = p by ;
hence x in the carrier of L ; :: thesis: verum
end;
hence A3: dom () = the carrier of L by XTUPLE_0:def 12; :: thesis: ( rng () = the carrier of L & field () = the carrier of L )
now :: thesis: for x being object holds
( ( x in the carrier of L implies ex y being object st [y,x] in LattRel L ) & ( ex y being object st [y,x] in LattRel L implies x in the carrier of L ) )
let x be object ; :: thesis: ( ( x in the carrier of L implies ex y being object st [y,x] in LattRel L ) & ( ex y being object st [y,x] in LattRel L implies x in the carrier of L ) )
thus ( x in the carrier of L implies ex y being object st [y,x] in LattRel L ) :: thesis: ( ex y being object st [y,x] in LattRel L implies x in the carrier of L )
proof
assume x in the carrier of L ; :: thesis: ex y being object st [y,x] in LattRel L
then reconsider p = x as Element of L ;
[p,p] in LattRel L ;
hence ex y being object st [y,x] in LattRel L ; :: thesis: verum
end;
given y being object such that A4: [y,x] in LattRel L ; :: thesis: x in the carrier of L
consider p, q being Element of L such that
A5: [y,x] = [p,q] and
p [= q by A4;
x = q by ;
hence x in the carrier of L ; :: thesis: verum
end;
hence rng () = the carrier of L by XTUPLE_0:def 13; :: thesis: field () = the carrier of L
hence field () = the carrier of L \/ the carrier of L by
.= the carrier of L ;
:: thesis: verum