let f, g be FinSequence of CQC-WFF ; :: thesis: ( len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & Ant f |= Suc f & Ant g |= Suc g implies Ant (Ant f) |= Suc f )
assume A1: ( len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & Ant f |= Suc f & Ant g |= Suc g ) ; :: thesis: Ant (Ant f) |= Suc f
then A2: ( len (Ant f) > 0 & len (Ant g) > 0 ) by Th4;
let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of A
for v being Element of Valuations_in A st J,v |= Ant (Ant f) holds
J,v |= Suc f
let J be interpretation of A; :: thesis: for v being Element of Valuations_in A st J,v |= Ant (Ant f) holds
J,v |= Suc f
let v be Element of Valuations_in A; :: thesis: ( J,v |= Ant (Ant f) implies J,v |= Suc f )
assume A3: J,v |= Ant (Ant f) ; :: thesis: J,v |= Suc f

A4: now

assume J,v |= Suc (Ant f) ; :: thesis: J,v |= Suc f
then J,v |= Ant f by A2, A3, Th17;
hence J,v |= Suc f by A1, Def15; :: thesis: verum

end;

now

assume not J,v |= Suc (Ant f) ; :: thesis: J,v |= Suc f
then J,v |= Suc (Ant g) by A1, VALUAT_1:28;
then J,v |= Ant g by A1, A2, A3, Th17;
hence J,v |= Suc f by A1, Def15; :: thesis: verum

end;

hence J,v |= Suc f by A4; :: thesis: verum