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

A1: now :: thesis: ( (Valid (p,J)) . v = TRUE implies ((Valid (p,J)) . v) '&' ('not' ((Valid (p,J)) . v)) = FALSE )

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 MARGREL1:11;
hence ((Valid (p,J)) . v) '&' ('not' ((Valid (p,J)) . v)) = FALSE by MARGREL1:12; :: thesis: verum

end;

A2: ( (Valid (p,J)) . v = FALSE implies ((Valid (p,J)) . v) '&' ('not' ((Valid (p,J)) . v)) = FALSE ) by MARGREL1:12;
(Valid ((p '&' ('not' p)),J)) . v = ((Valid (p,J)) . v) '&' ((Valid (('not' p),J)) . v) by Th12
.= ((Valid (p,J)) . v) '&' ('not' ((Valid (p,J)) . v)) by Th10 ;
hence (Valid ((p '&' ('not' p)),J)) . v = FALSE by A1, A2, XBOOLEAN:def 3; :: thesis: verum