let x be bound_QC-variable; :: thesis: for A being non empty set
for v being Element of Valuations_in A
for S being Element of CQC-Sub-WFF st x in dom (S `2 ) holds
(v . (Val_S v,S)) . x = (Val_S v,S) . x
let A be non empty set ; :: thesis: for v being Element of Valuations_in A
for S being Element of CQC-Sub-WFF st x in dom (S `2 ) holds
(v . (Val_S v,S)) . x = (Val_S v,S) . x
let v be Element of Valuations_in A; :: thesis: for S being Element of CQC-Sub-WFF st x in dom (S `2 ) holds
(v . (Val_S v,S)) . x = (Val_S v,S) . x
let S be Element of CQC-Sub-WFF ; :: thesis: ( x in dom (S `2 ) implies (v . (Val_S v,S)) . x = (Val_S v,S) . x )
assume x in dom (S `2 ) ; :: thesis: (v . (Val_S v,S)) . x = (Val_S v,S) . x
then A1: x in dom (@ (S `2 )) by SUBSTUT1:def 2;
( rng (@ (S `2 )) c= bound_QC-variables & dom v = bound_QC-variables ) by FUNCT_2:def 1;
then x in dom (Val_S v,S) by A1, RELAT_1:46;
hence (v . (Val_S v,S)) . x = (Val_S v,S) . x by FUNCT_4:14; :: thesis: verum