let A be non empty set ; :: thesis: for v being Element of Valuations_in A
for p being Element of CQC-WFF
for J being interpretation of A holds (Valid ((p '&' ('not' p)),J)) . v = FALSE
let v be Element of Valuations_in A; :: thesis: for p being Element of CQC-WFF
for J being interpretation of A holds (Valid ((p '&' ('not' p)),J)) . v = FALSE
let p be Element of CQC-WFF ; :: thesis: for J being interpretation of A holds (Valid ((p '&' ('not' p)),J)) . v = FALSE
let J be interpretation of A; :: thesis: (Valid ((p '&' ('not' p)),J)) . v = FALSE

A1: now

assume (Valid (p,J)) . v = TRUE ; :: thesis: ((Valid (p,J)) . v) '&' ('not' ((Valid (p,J)) . v)) = FALSE
then 'not' ((Valid (p,J)) . v) = FALSE by MARGREL1:41;
hence ((Valid (p,J)) . v) '&' ('not' ((Valid (p,J)) . v)) = FALSE by MARGREL1:45; :: thesis: verum

end;

A2: ( (Valid (p,J)) . v = FALSE implies ((Valid (p,J)) . v) '&' ('not' ((Valid (p,J)) . v)) = FALSE ) by MARGREL1:45;
(Valid ((p '&' ('not' p)),J)) . v = ((Valid (p,J)) . v) '&' ((Valid (('not' p),J)) . v) by Th22
.= ((Valid (p,J)) . v) '&' ('not' ((Valid (p,J)) . v)) by Th20 ;
hence (Valid ((p '&' ('not' p)),J)) . v = FALSE by A1, A2, XBOOLEAN:def 3; :: thesis: verum