reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;
reserve f for BinOp of D;
reserve a,a1,a2,b,b1,b2,A,B,C,X,Y,Z,x,x1,x2,y,y1,y2,z for set,
U,U1,U2,U3 for non empty set, u,u1,u2 for Element of U,
P,Q,R for Relation, f,f1,f2,g,g1,g2 for Function,
k,m,n for Nat, kk,mm,nn for Element of NAT, m1, n1 for non zero Nat,
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;

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;
