let A, B, C, D, E, F be set ; :: thesis: for h being Function
for A9, B9, C9, D9, E9, F9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds
( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 )
let h be Function; :: thesis: for A9, B9, C9, D9, E9, F9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds
( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 )
let A9, B9, C9, D9, E9, F9 be set ; :: thesis: ( A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) implies ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 ) )
assume that
A1: A <> B and
A2: A <> C and
A3: A <> D and
A4: A <> E and
A5: A <> F and
A6: ( B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) and
A7: h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) ; :: thesis: ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 )
A in dom (A .--> A9) by TARSKI:def 1;
then A9: h . A = (A .--> A9) . A by A7, FUNCT_4:13;
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)) . C by A7, FUNCT_4:11;
not F in dom (A .--> A9) by A5, TARSKI:def 1;
then A11: h . F = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . F by A7, FUNCT_4:11
.= F9 by A6, Th26 ;
not E in dom (A .--> A9) by A4, TARSKI:def 1;
then A12: h . E = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . E by A7, FUNCT_4:11
.= E9 by A6, Th26 ;
not D in dom (A .--> A9) by A3, TARSKI:def 1;
then A13: h . D = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . D by A7, FUNCT_4:11
.= D9 by A6, Th26 ;
not B in dom (A .--> A9) by A1, TARSKI:def 1;
then h . B = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . B by A7, FUNCT_4:11
.= B9 by A6, Th26 ;
hence ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 ) by A6, A9, A10, A13, A12, A11, Th26, FUNCOP_1:72; :: thesis: verum