let Al be QC-alphabet ; :: thesis: for A being non empty set
for p being Element of CQC-WFF Al
for J being interpretation of Al,A holds Valid ((p '&' p),J) = Valid (p,J)
let A be non empty set ; :: thesis: for p being Element of CQC-WFF Al
for J being interpretation of Al,A holds Valid ((p '&' p),J) = Valid (p,J)
let p be Element of CQC-WFF Al; :: thesis: for J being interpretation of Al,A holds Valid ((p '&' p),J) = Valid (p,J)
let J be interpretation of Al,A; :: thesis: Valid ((p '&' p),J) = Valid (p,J)

now :: thesis: for v being Element of Valuations_in (Al,A) holds (Valid ((p '&' p),J)) . v = (Valid (p,J)) . v

let v be Element of Valuations_in (Al,A); :: thesis: (Valid ((p '&' p),J)) . v = (Valid (p,J)) . v
thus (Valid ((p '&' p),J)) . v = ((Valid (p,J)) . v) '&' ((Valid (p,J)) . v) by Th12
.= (Valid (p,J)) . v ; :: thesis: verum

end;

hence Valid ((p '&' p),J) = Valid (p,J) by FUNCT_2:63; :: thesis: verum