theorem (ex p st x=p & p is X*-valued) implies U-multiCat.x is X-valued
proof
set C=U-multiCat;
A1: dom C=U** by FUNCT_2:def 1; given p such that
A2: x=p & p is X*-valued; x is FinSequence of X* by A2, Lm1;
then reconsider px=x as Element of X**;
per cases;
suppose
A3: C.p<>{}; then p in U** & p<>{} by FUNCT_1:def 2, A1; then
reconsider pp=x as non empty FinSequence of U* by Lm1, A2;
A4: pp is X*-valued & not pp is {}*-valued by Th52, A2, A3;
reconsider XX=X as non empty set by Th52, A2, A3; set CX=XX-multiCat;
reconsider pxx=px as Element of XX**; CX.pp<>{} by Th52, A4;
hence thesis by Th52, A3, A2;
end;
suppose C.p={}; then reconsider e=C.p as empty set;
rng e c= X; hence thesis by  A2;
end;
end;
