let Al be QC-alphabet ; for x being bound_QC-variable of Al
for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let x be bound_QC-variable of Al; for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let A be non empty set ; for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let J be interpretation of Al,A; for v being Element of Valuations_in (Al,A)
for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let v be Element of Valuations_in (Al,A); for S being Element of CQC-Sub-WFF Al
for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let S be Element of CQC-Sub-WFF Al; for xSQ being second_Q_comp of [S,x] holds
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
let xSQ be second_Q_comp of [S,x]; ( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) iff for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
thus
( ( for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S ) implies for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S )
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) implies for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )proof
assume A1:
for
a being
Element of
A holds
J,
v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
;
for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
let a be
Element of
A;
J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) = (v . (NEx_Val (v,S,x,xSQ))) . (x | a)
by FUNCT_4:14;
hence
J,
(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
by A1;
verum
end;
thus
( ( for a being Element of A holds J,(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S ) implies for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S )
verumproof
assume A2:
for
a being
Element of
A holds
J,
(v . (NEx_Val (v,S,x,xSQ))) . (x | a) |= S
;
for a being Element of A holds J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
let a be
Element of
A;
J,v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) = (v . (NEx_Val (v,S,x,xSQ))) . (x | a)
by FUNCT_4:14;
hence
J,
v . ((NEx_Val (v,S,x,xSQ)) +* (x | a)) |= S
by A2;
verum
end;