let Al be QC-alphabet ; 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; 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 ; 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); 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; ( x in dom (S `2) implies (v . (Val_S (v,S))) . x = (Val_S (v,S)) . x )
assume
x in dom (S `2)
; (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; verum