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 x in dom (S `2) holds
(v . (Val_S (v,S))) . x = (Val_S (v,S)) . 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 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 (Al,A)
for S being Element of CQC-Sub-WFF Al 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 (Al,A); :: thesis: for S being Element of CQC-Sub-WFF Al 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 Al; :: 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 Al & dom v = bound_QC-variables Al ) by FUNCT_2:def 1;
then x in dom (Val_S (v,S)) by A1, RELAT_1:27;
hence (v . (Val_S (v,S))) . x = (Val_S (v,S)) . x by FUNCT_4:13; :: thesis: verum