let S be non empty non void 4,1 integer BoolSignature ; :: thesis: for I being integer SortSymbol of S holds

( I <> the bool-sort of S & the connectives of S . 4 is_of_type {} ,I & the connectives of S . (4 + 1) is_of_type {} ,I & the connectives of S . 4 <> the connectives of S . (4 + 1) & the connectives of S . (4 + 2) is_of_type <*I*>,I & the connectives of S . (4 + 3) is_of_type <*I,I*>,I & the connectives of S . (4 + 4) is_of_type <*I,I*>,I & the connectives of S . (4 + 5) is_of_type <*I,I*>,I & the connectives of S . (4 + 3) <> the connectives of S . (4 + 4) & the connectives of S . (4 + 3) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 4) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 6) is_of_type <*I,I*>, the bool-sort of S )

let I be integer SortSymbol of S; :: thesis: ( I <> the bool-sort of S & the connectives of S . 4 is_of_type {} ,I & the connectives of S . (4 + 1) is_of_type {} ,I & the connectives of S . 4 <> the connectives of S . (4 + 1) & the connectives of S . (4 + 2) is_of_type <*I*>,I & the connectives of S . (4 + 3) is_of_type <*I,I*>,I & the connectives of S . (4 + 4) is_of_type <*I,I*>,I & the connectives of S . (4 + 5) is_of_type <*I,I*>,I & the connectives of S . (4 + 3) <> the connectives of S . (4 + 4) & the connectives of S . (4 + 3) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 4) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 6) is_of_type <*I,I*>, the bool-sort of S )

A1: I = 1 by Def39;

ex I being SortSymbol of S st

( I = 1 & I <> the bool-sort of S & the connectives of S . 4 is_of_type {} ,I & the connectives of S . (4 + 1) is_of_type {} ,I & the connectives of S . 4 <> the connectives of S . (4 + 1) & the connectives of S . (4 + 2) is_of_type <*I*>,I & the connectives of S . (4 + 3) is_of_type <*I,I*>,I & the connectives of S . (4 + 4) is_of_type <*I,I*>,I & the connectives of S . (4 + 5) is_of_type <*I,I*>,I & the connectives of S . (4 + 3) <> the connectives of S . (4 + 4) & the connectives of S . (4 + 3) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 4) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 6) is_of_type <*I,I*>, the bool-sort of S ) by Def38;

hence ( I <> the bool-sort of S & the connectives of S . 4 is_of_type {} ,I & the connectives of S . (4 + 1) is_of_type {} ,I & the connectives of S . 4 <> the connectives of S . (4 + 1) & the connectives of S . (4 + 2) is_of_type <*I*>,I & the connectives of S . (4 + 3) is_of_type <*I,I*>,I & the connectives of S . (4 + 4) is_of_type <*I,I*>,I & the connectives of S . (4 + 5) is_of_type <*I,I*>,I & the connectives of S . (4 + 3) <> the connectives of S . (4 + 4) & the connectives of S . (4 + 3) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 4) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 6) is_of_type <*I,I*>, the bool-sort of S ) by A1; :: thesis: verum

( I <> the bool-sort of S & the connectives of S . 4 is_of_type {} ,I & the connectives of S . (4 + 1) is_of_type {} ,I & the connectives of S . 4 <> the connectives of S . (4 + 1) & the connectives of S . (4 + 2) is_of_type <*I*>,I & the connectives of S . (4 + 3) is_of_type <*I,I*>,I & the connectives of S . (4 + 4) is_of_type <*I,I*>,I & the connectives of S . (4 + 5) is_of_type <*I,I*>,I & the connectives of S . (4 + 3) <> the connectives of S . (4 + 4) & the connectives of S . (4 + 3) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 4) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 6) is_of_type <*I,I*>, the bool-sort of S )

let I be integer SortSymbol of S; :: thesis: ( I <> the bool-sort of S & the connectives of S . 4 is_of_type {} ,I & the connectives of S . (4 + 1) is_of_type {} ,I & the connectives of S . 4 <> the connectives of S . (4 + 1) & the connectives of S . (4 + 2) is_of_type <*I*>,I & the connectives of S . (4 + 3) is_of_type <*I,I*>,I & the connectives of S . (4 + 4) is_of_type <*I,I*>,I & the connectives of S . (4 + 5) is_of_type <*I,I*>,I & the connectives of S . (4 + 3) <> the connectives of S . (4 + 4) & the connectives of S . (4 + 3) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 4) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 6) is_of_type <*I,I*>, the bool-sort of S )

A1: I = 1 by Def39;

ex I being SortSymbol of S st

( I = 1 & I <> the bool-sort of S & the connectives of S . 4 is_of_type {} ,I & the connectives of S . (4 + 1) is_of_type {} ,I & the connectives of S . 4 <> the connectives of S . (4 + 1) & the connectives of S . (4 + 2) is_of_type <*I*>,I & the connectives of S . (4 + 3) is_of_type <*I,I*>,I & the connectives of S . (4 + 4) is_of_type <*I,I*>,I & the connectives of S . (4 + 5) is_of_type <*I,I*>,I & the connectives of S . (4 + 3) <> the connectives of S . (4 + 4) & the connectives of S . (4 + 3) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 4) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 6) is_of_type <*I,I*>, the bool-sort of S ) by Def38;

hence ( I <> the bool-sort of S & the connectives of S . 4 is_of_type {} ,I & the connectives of S . (4 + 1) is_of_type {} ,I & the connectives of S . 4 <> the connectives of S . (4 + 1) & the connectives of S . (4 + 2) is_of_type <*I*>,I & the connectives of S . (4 + 3) is_of_type <*I,I*>,I & the connectives of S . (4 + 4) is_of_type <*I,I*>,I & the connectives of S . (4 + 5) is_of_type <*I,I*>,I & the connectives of S . (4 + 3) <> the connectives of S . (4 + 4) & the connectives of S . (4 + 3) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 4) <> the connectives of S . (4 + 5) & the connectives of S . (4 + 6) is_of_type <*I,I*>, the bool-sort of S ) by A1; :: thesis: verum