let P, Q be pcs-Str ; :: thesis: for p, q being Element of holds
( p <= q iff ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ) )
set R = pcs-sum P,Q;
let p, q be Element of ; :: thesis: ( p <= q iff ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ) )
A1: the InternalRel of (pcs-sum P,Q) = the InternalRel of P \/ the InternalRel of Q by Th14;
thus ( not p <= q or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ) :: thesis: ( ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ) implies p <= q )

proof

assume A2: [p,q] in the InternalRel of (pcs-sum P,Q) ; :: according to ORDERS_2:def 9 :: thesis: ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) )

per cases ( [p,q] in the InternalRel of P or [p,q] in the InternalRel of Q ) by A1, A2, XBOOLE_0:def 3;

suppose A3: [p,q] in the InternalRel of P ; :: thesis: ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) )

then reconsider p' = p, q' = q as Element of by ZFMISC_1:106;
p' <= q' by A3, ORDERS_2:def 9;
hence ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ) ; :: thesis: verum

end;

suppose A4: [p,q] in the InternalRel of Q ; :: thesis: ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) )

then reconsider p' = p, q' = q as Element of by ZFMISC_1:106;
p' <= q' by A4, ORDERS_2:def 9;
hence ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ) ; :: thesis: verum

end;

end;

end;

assume A5: ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ) ; :: thesis: p <= q

per cases ( ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) or ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ) by A5;

suppose ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ; :: thesis: p <= q

then consider p', q' being Element of such that
A6: p' = p and
A7: q' = q and
A8: p' <= q' ;
[p',q'] in the InternalRel of P by A8, ORDERS_2:def 9;
hence [p,q] in the InternalRel of (pcs-sum P,Q) by A1, A6, A7, XBOOLE_0:def 3; :: according to ORDERS_2:def 9 :: thesis: verum

end;

suppose ex p', q' being Element of st
( p' = p & q' = q & p' <= q' ) ; :: thesis: p <= q

then consider p', q' being Element of such that
A9: p' = p and
A10: q' = q and
A11: p' <= q' ;
[p',q'] in the InternalRel of Q by A11, ORDERS_2:def 9;
hence [p,q] in the InternalRel of (pcs-sum P,Q) by A1, A9, A10, XBOOLE_0:def 3; :: according to ORDERS_2:def 9 :: thesis: verum

end;

end;