let A be non-empty MSAlgebra over S; :: thesis: ( A is finite-yielding implies A is finitely-generated )
assume A1: A is finite-yielding ; :: thesis: A 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 = A holds
ex G being GeneratorSet of A st G is finite-yielding

let S9 be non empty non void ManySortedSign ; :: thesis: ( S9 = S implies for A being MSAlgebra over S9 st A = A holds
ex G being GeneratorSet of A st G is finite-yielding )
assume A2: S9 = S ; :: thesis: for A being MSAlgebra over S9 st A = A holds
ex G being GeneratorSet of A st G is finite-yielding
let A9 be MSAlgebra over S9; :: thesis: ( A9 = A implies ex G being GeneratorSet of A9 st G is finite-yielding )
assume A9 = A ; :: thesis: not for G being GeneratorSet of A9 holds G is finite-yielding
then reconsider G = the Sorts of A as GeneratorSet of A9 by A2, Th6;
take G ; :: thesis: G is finite-yielding
thus G is finite-yielding by A1; :: thesis: verum

end;

case S is void ; :: thesis: the Sorts of A is finite-yielding

thus the Sorts of A is finite-yielding by A1; :: thesis: verum

end;

end;