let S be non empty non void ManySortedSign ; :: thesis: for X being non-empty finite-yielding ManySortedSet of the carrier of S holds FreeMSA X is finitely-generated
let X be non-empty finite-yielding ManySortedSet of the carrier of S; :: thesis: FreeMSA X is finitely-generated

per cases ( not S is void or S is void ) ;
:: according to MSAFREE2:def 10

case not S is void ; :: thesis: for S9 being non empty non void ManySortedSign st S9 = S holds
for A being MSAlgebra over S9 st A = FreeMSA X holds
ex G being GeneratorSet of A st G is finite-yielding

reconsider G = FreeGen X as GeneratorSet of FreeMSA X ;
let S9 be non empty non void ManySortedSign ; :: thesis: ( S9 = S implies for A being MSAlgebra over S9 st A = FreeMSA X holds
ex G being GeneratorSet of A st G is finite-yielding )
assume A1: S9 = S ; :: thesis: for A being MSAlgebra over S9 st A = FreeMSA X holds
ex G being GeneratorSet of A st G is finite-yielding
let A be MSAlgebra over S9; :: thesis: ( A = FreeMSA X implies ex G being GeneratorSet of A st G is finite-yielding )
assume A = FreeMSA X ; :: thesis: not for G being GeneratorSet of A holds G is finite-yielding
then reconsider G = G as GeneratorSet of A by A1;
take G ; :: thesis: G is finite-yielding
thus G is finite-yielding :: thesis: verum

proof

let i be object ; :: according to FINSET_1:def 5 :: thesis: ( not i in the carrier of S9 or G . i is finite )
reconsider Gi = G . i as set ;
assume i in the carrier of S9 ; :: thesis: G . i is finite
then reconsider iS = i as SortSymbol of S by A1;
reconsider Xi = X . iS as non empty finite set ;

now :: thesis: ex f, f being Function st
( f is one-to-one & dom f = Gi & rng f c= Xi )

defpred S₁[ object , object ] means $1 = root-tree [$2,i];
A2: for e being object st e in Gi holds
ex u being object st
( u in Xi & S₁[e,u] )

proof

A3: Gi = FreeGen (iS,X) by MSAFREE:def 16;
let e be object ; :: thesis: ( e in Gi implies ex u being object st
( u in Xi & S₁[e,u] ) )
assume e in Gi ; :: thesis: ex u being object st
( u in Xi & S₁[e,u] )
then consider u being set such that
A4: ( u in Xi & e = root-tree [u,i] ) by A3, MSAFREE:def 15;
take u ; :: thesis: ( u in Xi & S₁[e,u] )
thus ( u in Xi & S₁[e,u] ) by A4; :: thesis: verum

end;

consider f being Function such that
A5: dom f = Gi and
A6: rng f c= Xi and
A7: for e being object st e in Gi holds
S₁[e,f . e] from FUNCT_1:sch 6(A2);
take f = f; :: thesis: ex f being Function st
( f is one-to-one & dom f = Gi & rng f c= Xi )
f is one-to-one

proof

let x1, x2 be object ; :: according to FUNCT_1:def 4 :: thesis: ( not x1 in dom f or not x2 in dom f or not f . x1 = f . x2 or x1 = x2 )
assume that
A8: x1 in dom f and
A9: x2 in dom f and
A10: f . x1 = f . x2 ; :: thesis: x1 = x2
thus x1 = root-tree [(f . x2),i] by A5, A7, A8, A10
.= x2 by A5, A7, A9 ; :: thesis: verum

end;

hence ex f being Function st
( f is one-to-one & dom f = Gi & rng f c= Xi ) by A5, A6; :: thesis: verum

end;

then ( card Gi c= card Xi or card Gi in card Xi ) by CARD_1:10;
hence G . i is finite by CARD_2:49; :: thesis: verum

end;

end;

case S is void ; :: thesis: the Sorts of (FreeMSA X) is finite-yielding

hence the Sorts of (FreeMSA X) is finite-yielding ; :: thesis: verum

end;

end;