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 . ((@ (S `2 )) . x) = (v . (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 . ((@ (S `2 )) . x) = (v . (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 . ((@ (S `2 )) . x) = (v . (Val_S v,S)) . x
let S be Element of CQC-Sub-WFF ; :: thesis: ( x in dom (S `2 ) implies v . ((@ (S `2 )) . x) = (v . (Val_S v,S)) . x )
assume A1: x in dom (S `2 ) ; :: thesis: v . ((@ (S `2 )) . x) = (v . (Val_S v,S)) . x
then A2: (v . (Val_S v,S)) . x = (Val_S v,S) . x by Th13;
x in dom (@ (S `2 )) by A1, SUBSTUT1:def 2;
hence v . ((@ (S `2 )) . x) = (v . (Val_S v,S)) . x by A2, FUNCT_1:23; :: thesis: verum