let A, B, C, D, E, F, J, M be set ; :: thesis: for h being Function
for A9, B9, C9, D9, E9, F9, J9, M9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds
( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 )
let h be Function; :: thesis: for A9, B9, C9, D9, E9, F9, J9, M9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds
( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 )
let A9, B9, C9, D9, E9, F9, J9, M9 be set ; :: thesis: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) implies ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) )
assume that
A1: A <> B and
A2: A <> C and
A3: A <> D and
A4: A <> E and
A5: A <> F and
A6: A <> J and
A7: ( B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M ) and
A8: h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) ; :: thesis: ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 )
not C in dom (A .--> A9) by A2, TARSKI:def 1;
then A10: h . C = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . C by A8, FUNCT_4:11;
not J in dom (A .--> A9) by A6, TARSKI:def 1;
then A11: h . J = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . J by A8, FUNCT_4:11
.= J9 by A7, Th49 ;
not F in dom (A .--> A9) by A5, TARSKI:def 1;
then A12: h . F = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . F by A8, FUNCT_4:11
.= F9 by A7, Th49 ;
not E in dom (A .--> A9) by A4, TARSKI:def 1;
then A13: h . E = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . E by A8, FUNCT_4:11
.= E9 by A7, Th49 ;
not D in dom (A .--> A9) by A3, TARSKI:def 1;
then A14: h . D = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . D by A8, FUNCT_4:11
.= D9 by A7, Th49 ;
not B in dom (A .--> A9) by A1, TARSKI:def 1;
then h . B = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . B by A8, FUNCT_4:11
.= B9 by A7, Th49 ;
hence ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) by A7, A10, A14, A13, A12, A11, Th49; :: thesis: verum