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 Th58: not y in proj2 X implies [:A,{y}:] misses X
proof
set X2=proj2 X, Y=[:A,{y}:], Z=X/\Y; assume A1: not y in X2; assume
Y meets X; then Z<>{};
then consider z being object such that
A2: z in Z; set x1=z`1, y1=z`2;
x1 in A & y1 in {y} & z=[x1,y1] & z in X by A2, MCART_1:10,21; then
y1=y & y1 in X2 by TARSKI:def 1, XTUPLE_0:def 13;
hence contradiction by A1;
end;
