let P be pcs-Str ; for a being set
for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <= q holds
p1 <= q1
let a be set ; for p, q being Element of P
for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <= q holds
p1 <= q1
let p, q be Element of P; for p1, q1 being Element of (pcs-extension (P,a)) st p = p1 & q = q1 & p <= q holds
p1 <= q1
let p1, q1 be Element of (pcs-extension (P,a)); ( p = p1 & q = q1 & p <= q implies p1 <= q1 )
assume that
A1:
p = p1
and
A2:
q = q1
and
A3:
[p,q] in the InternalRel of P
; ORDERS_2:def 5 p1 <= q1
the InternalRel of P c= the InternalRel of (pcs-extension (P,a))
by Th21;
hence
[p1,q1] in the InternalRel of (pcs-extension (P,a))
by A1, A2, A3; ORDERS_2:def 5 verum