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 not x in dom (S `2 ) holds
(v . (Val_S v,S)) . x = v . 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 not x in dom (S `2 ) holds
(v . (Val_S v,S)) . x = v . x
let v be Element of Valuations_in A; :: thesis: for S being Element of CQC-Sub-WFF st not x in dom (S `2 ) holds
(v . (Val_S v,S)) . x = v . x
let S be Element of CQC-Sub-WFF ; :: thesis: ( not x in dom (S `2 ) implies (v . (Val_S v,S)) . x = v . x )
assume not x in dom (S `2 ) ; :: thesis: (v . (Val_S v,S)) . x = v . x
then A1: not x in dom (@ (S `2 )) by SUBSTUT1:def 2;
dom ((@ (S `2 )) * v) c= dom (@ (S `2 )) by RELAT_1:44;
then not x in dom (Val_S v,S) by A1;
hence (v . (Val_S v,S)) . x = v . x by FUNCT_4:12; :: thesis: verum