let x be object ; :: thesis: for D being non empty set
for p, q being PartialPredicate of D st x in dom p & p . x = FALSE holds
(PP_and (p,q)) . x = FALSE
let D be non empty set ; :: thesis: for p, q being PartialPredicate of D st x in dom p & p . x = FALSE holds
(PP_and (p,q)) . x = FALSE
let p, q be PartialPredicate of D; :: thesis: ( x in dom p & p . x = FALSE implies (PP_and (p,q)) . x = FALSE )
assume that
A1: x in dom p and
A2: p . x = FALSE ; :: thesis: (PP_and (p,q)) . x = FALSE
A3: (PP_not p) . x = TRUE by A1, A2, Def2;
A4: dom (PP_not p) = dom p by Def2;
A5: dom (PP_or ((PP_not p),(PP_not q))) = { d where d is Element of D : ( ( d in dom (PP_not p) & (PP_not p) . d = TRUE ) or ( d in dom (PP_not q) & (PP_not q) . d = TRUE ) or ( d in dom (PP_not p) & (PP_not p) . d = FALSE & d in dom (PP_not q) & (PP_not q) . d = FALSE ) ) } by Def4;
A6: x in dom (PP_or ((PP_not p),(PP_not q))) by A1, A3, A4, A5;
(PP_or ((PP_not p),(PP_not q))) . x = TRUE by A1, A3, A4, Def4;
hence (PP_and (p,q)) . x = FALSE by A6, Def2; :: thesis: verum