let A be MP-wff; :: thesis: ( A = VERUM or ex p being MP-variable st A = @ p or ex B being MP-wff st A = 'not' B or ex B being MP-wff st A = (#) B or ex B, C being MP-wff st A = B '&' C )

now

per cases ( card (dom A) = 1 or card (dom A) >= 2 ) by NAT_1:26;

suppose card (dom A) = 1 ; :: thesis: ( A = VERUM or ex p being MP-variable st A = @ p or ex B being MP-wff st A = 'not' B or ex B being MP-wff st A = (#) B or ex B, C being MP-wff st A = B '&' C )

hence ( A = VERUM or ex p being MP-variable st A = @ p or ex B being MP-wff st A = 'not' B or ex B being MP-wff st A = (#) B or ex B, C being MP-wff st A = B '&' C ) by Th42; :: thesis: verum

end;

suppose card (dom A) >= 2 ; :: thesis: ( A = VERUM or ex p being MP-variable st A = @ p or ex B being MP-wff st A = 'not' B or ex B being MP-wff st A = (#) B or ex B, C being MP-wff st A = B '&' C )

hence ( A = VERUM or ex p being MP-variable st A = @ p or ex B being MP-wff st A = 'not' B or ex B being MP-wff st A = (#) B or ex B, C being MP-wff st A = B '&' C ) by Th43; :: thesis: verum

end;

end;

end;

hence ( A = VERUM or ex p being MP-variable st A = @ p or ex B being MP-wff st A = 'not' B or ex B being MP-wff st A = (#) B or ex B, C being MP-wff st A = B '&' C ) ; :: thesis: verum