let A, B, C, D, E, F, J, M, N be set ; :: thesis: for h being Function
for A', B', C', D', E', F', J', M', N' being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & h = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) +* (A .--> A') holds
( h . A = A' & h . B = B' & h . C = C' & h . D = D' & h . E = E' & h . F = F' & h . J = J' & h . M = M' & h . N = N' )
let h be Function; :: thesis: for A', B', C', D', E', F', J', M', N' being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & h = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) +* (A .--> A') holds
( h . A = A' & h . B = B' & h . C = C' & h . D = D' & h . E = E' & h . F = F' & h . J = J' & h . M = M' & h . N = N' )
let A', B', C', D', E', F', J', M', N' be set ; :: thesis: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & h = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) +* (A .--> A') implies ( h . A = A' & h . B = B' & h . C = C' & h . D = D' & h . E = E' & h . F = F' & h . J = J' & h . M = M' & h . N = N' ) )
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: A <> M and
A8: A <> N and
A9: ( B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N ) and
A10: M <> N and
A11: h = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) +* (A .--> A') ; :: thesis: ( h . A = A' & h . B = B' & h . C = C' & h . D = D' & h . E = E' & h . F = F' & h . J = J' & h . M = M' & h . N = N' )
A12: dom (A .--> A') = {A} by FUNCOP_1:19;
then A in dom (A .--> A') by TARSKI:def 1;
then A13: h . A = (A .--> A') . A by A11, FUNCT_4:14;
not E in dom (A .--> A') by A4, A12, TARSKI:def 1;
then A14: h . E = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) . E by A11, FUNCT_4:12
.= E' by A9, Th65 ;
not N in dom (A .--> A') by A8, A12, TARSKI:def 1;
then A15: h . N = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) . N by A11, FUNCT_4:12
.= N' by FUNCT_7:96 ;
not D in dom (A .--> A') by A3, A12, TARSKI:def 1;
then A16: h . D = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) . D by A11, FUNCT_4:12
.= D' by A9, Th65 ;
not C in dom (A .--> A') by A2, A12, TARSKI:def 1;
then A17: h . C = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) . C by A11, FUNCT_4:12;
not J in dom (A .--> A') by A6, A12, TARSKI:def 1;
then A18: h . J = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) . J by A11, FUNCT_4:12
.= J' by A9, Th65 ;
not F in dom (A .--> A') by A5, A12, TARSKI:def 1;
then A19: h . F = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) . F by A11, FUNCT_4:12
.= F' by A9, Th65 ;
not M in dom (A .--> A') by A7, A12, TARSKI:def 1;
then A20: h . M = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) . M by A11, FUNCT_4:12
.= M' by A10, Lm2 ;
not B in dom (A .--> A') by A1, A12, TARSKI:def 1;
then h . B = ((((((((B .--> B') +* (C .--> C')) +* (D .--> D')) +* (E .--> E')) +* (F .--> F')) +* (J .--> J')) +* (M .--> M')) +* (N .--> N')) . B by A11, FUNCT_4:12
.= B' by A9, Th65 ;
hence ( h . A = A' & h . B = B' & h . C = C' & h . D = D' & h . E = E' & h . F = F' & h . J = J' & h . M = M' & h . N = N' ) by A9, A13, A17, A16, A14, A19, A18, A20, A15, Th65, FUNCOP_1:87; :: thesis: verum