let Al be QC-alphabet ; :: thesis: for A being non empty set
for v being Element of Valuations_in (Al,A)
for p, q being Element of CQC-WFF Al
for J being interpretation of Al,A holds J,v |= (p '&' q) => (q '&' p)
let A be non empty set ; :: thesis: for v being Element of Valuations_in (Al,A)
for p, q being Element of CQC-WFF Al
for J being interpretation of Al,A holds J,v |= (p '&' q) => (q '&' p)
let v be Element of Valuations_in (Al,A); :: thesis: for p, q being Element of CQC-WFF Al
for J being interpretation of Al,A holds J,v |= (p '&' q) => (q '&' p)
let p, q be Element of CQC-WFF Al; :: thesis: for J being interpretation of Al,A holds J,v |= (p '&' q) => (q '&' p)
let J be interpretation of Al,A; :: thesis: J,v |= (p '&' q) => (q '&' p)
thus (Valid (((p '&' q) => (q '&' p)),J)) . v = TRUE :: according to VALUAT_1:def 7 :: thesis: verum

proof

assume not (Valid (((p '&' q) => (q '&' p)),J)) . v = TRUE ; :: thesis: contradiction
then A1: (Valid (((p '&' q) => (q '&' p)),J)) . v = FALSE by XBOOLEAN:def 3;
(Valid (((p '&' q) => (q '&' p)),J)) . v = (Valid (('not' ((p '&' q) '&' ('not' (q '&' p)))),J)) . v by QC_LANG2:def 2
.= 'not' ((Valid (((p '&' q) '&' ('not' (q '&' p))),J)) . v) by Th10
.= 'not' (((Valid ((p '&' q),J)) . v) '&' ((Valid (('not' (q '&' p)),J)) . v)) by Th12
.= 'not' (((Valid ((p '&' q),J)) . v) '&' ('not' ((Valid ((q '&' p),J)) . v))) by Th10 ;
then A2: ((Valid ((p '&' q),J)) . v) '&' ('not' ((Valid ((q '&' p),J)) . v)) = TRUE by A1, MARGREL1:11;
then 'not' ((Valid ((q '&' p),J)) . v) = TRUE by MARGREL1:12;
then A3: (Valid ((q '&' p),J)) . v = FALSE by MARGREL1:11;
(Valid ((p '&' q),J)) . v = TRUE by A2, MARGREL1:12;
then ((Valid (p,J)) . v) '&' ((Valid (q,J)) . v) = TRUE by Th12;
hence contradiction by A3, Th12; :: thesis: verum

end;