let F, G be Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)); :: thesis: ( ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
F . (t,h) = p '&' q ) & ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
G . (t,h) = p '&' q ) implies F = G )
assume A7: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
F . (t,h) = p '&' q ; :: thesis: ( ex t being QC-symbol of A ex h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) ex p, q being Element of CQC-WFF A st
( p = f . (t,h) & q = g . ((t + n),h) & not G . (t,h) = p '&' q ) or F = G )
assume A8: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
G . (t,h) = p '&' q ; :: thesis: F = G
for a being Element of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds F . a = G . a
proof
let a be Element of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]; :: thesis: F . a = G . a
consider k being Element of QC-symbols A, h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) such that
A9: a = [k,h] by DOMAIN_1:1;
reconsider q = g . ((k + n),h) as Element of CQC-WFF A ;
reconsider p = f . (k,h) as Element of CQC-WFF A ;
F . (k,h) = p '&' q by A7
.= G . (k,h) by A8 ;
hence F . a = G . a by A9; :: thesis: verum
end;
hence F = G by FUNCT_2:63; :: thesis: verum