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 S being Element of CQC-Sub-WFF Al st not x in dom (S `2) holds
(v . (Val_S (v,S))) . x = v . 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 S being Element of CQC-Sub-WFF Al 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 (Al,A)
for S being Element of CQC-Sub-WFF Al st not x in dom (S `2) holds
(v . (Val_S (v,S))) . x = v . x
let v be Element of Valuations_in (Al,A); :: thesis: for S being Element of CQC-Sub-WFF Al st not x in dom (S `2) holds
(v . (Val_S (v,S))) . x = v . x
let S be Element of CQC-Sub-WFF Al; :: 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_1:25;
then not x in dom (Val_S (v,S)) by A1;
hence (v . (Val_S (v,S))) . x = v . x by FUNCT_4:11; :: thesis: verum