let Al be QC-alphabet ; :: thesis: for x being bound_QC-variable of Al
for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let x be bound_QC-variable of Al; :: thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let A be non empty set ; :: thesis: for v being Element of Valuations_in (Al,A)
for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let v be Element of Valuations_in (Al,A); :: thesis: for a being Element of A
for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let a be Element of A; :: thesis: for X being set st X c= bound_QC-variables Al holds
( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
let X be set ; :: thesis: ( X c= bound_QC-variables Al implies ( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X ) )
A1: dom ((v . (x | a)) | X) = (dom (v . (x | a))) /\ X by RELAT_1:61;
dom (v | X) = (dom v) /\ X by RELAT_1:61;
then A2: dom (v | X) = (bound_QC-variables Al) /\ X by Th58;
assume X c= bound_QC-variables Al ; :: thesis: ( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X )
hence ( dom (v | X) = dom ((v . (x | a)) | X) & dom (v | X) = X ) by A2, A1, Th58, XBOOLE_1:28; :: thesis: verum