A2: for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set
for n being Nat holds F5(S,A,x,n) is non-empty MSAlgebra over F4(S,x,n) by A1;
consider f, g, h being ManySortedSet of NAT such that
A3: ( f . 0 = F1() & g . 0 = F2() & h . 0 = F3() ) and
A4: for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = F4(S,x,n) & g . (n + 1) = F5(S,A,x,n) & h . (n + 1) = F6(x,n) ) from A5: for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n & S2[S,A,x,n] holds
S2[F4(S,x,n),F5(S,A,x,n),F6(x,n),n + 1] ;
A6: ex S being non empty ManySortedSign ex A being non-empty MSAlgebra over S ex x being set st
( S = f . 0 & A = g . 0 & x = h . 0 & S2[S,A,x, 0 ] ) by A3;
for n being Nat ex S being non empty ManySortedSign ex A being non-empty MSAlgebra over S st
( S = f . n & A = g . n & S2[S,A,h . n,n] ) from CIRCCMB2:sch 13(A6, A4, A5, A2);
then consider S being non empty ManySortedSign , A being non-empty MSAlgebra over S such that
A7: ( S = f . F7() & A = g . F7() ) ;
take S ; :: thesis: ex A being non-empty MSAlgebra over S ex f, g, h being ManySortedSet of NAT st
( S = f . F7() & A = g . F7() & f . 0 = F1() & g . 0 = F2() & h . 0 = F3() & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = F4(S,x,n) & g . (n + 1) = F5(S,A,x,n) & h . (n + 1) = F6(x,n) ) ) )

take A ; :: thesis: ex f, g, h being ManySortedSet of NAT st
( S = f . F7() & A = g . F7() & f . 0 = F1() & g . 0 = F2() & h . 0 = F3() & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = F4(S,x,n) & g . (n + 1) = F5(S,A,x,n) & h . (n + 1) = F6(x,n) ) ) )

take f ; :: thesis: ex g, h being ManySortedSet of NAT st
( S = f . F7() & A = g . F7() & f . 0 = F1() & g . 0 = F2() & h . 0 = F3() & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = F4(S,x,n) & g . (n + 1) = F5(S,A,x,n) & h . (n + 1) = F6(x,n) ) ) )

take g ; :: thesis: ex h being ManySortedSet of NAT st
( S = f . F7() & A = g . F7() & f . 0 = F1() & g . 0 = F2() & h . 0 = F3() & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = F4(S,x,n) & g . (n + 1) = F5(S,A,x,n) & h . (n + 1) = F6(x,n) ) ) )

take h ; :: thesis: ( S = f . F7() & A = g . F7() & f . 0 = F1() & g . 0 = F2() & h . 0 = F3() & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = F4(S,x,n) & g . (n + 1) = F5(S,A,x,n) & h . (n + 1) = F6(x,n) ) ) )

thus ( S = f . F7() & A = g . F7() & f . 0 = F1() & g . 0 = F2() & h . 0 = F3() & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for x being set st S = f . n & A = g . n & x = h . n holds
( f . (n + 1) = F4(S,x,n) & g . (n + 1) = F5(S,A,x,n) & h . (n + 1) = F6(x,n) ) ) ) by A3, A4, A7; :: thesis: verum