reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem Th18:
  for PA,PC being a_partition of Y st
  PA '<' PC & x in PC & z0 in PA & t in x & t in z0 holds z0 c= x
proof
  let PA,PC be a_partition of Y;
  assume that
A1: PA '<' PC and
A2: x in PC and
A3: z0 in PA and
A4: t in x & t in z0;
  consider b such that
A5: b in PC and
A6: z0 c= b by A1,A3,SETFAM_1:def 2;
 x = b or x misses b by A2,A5,EQREL_1:def 4;
  hence thesis by A4,A6,XBOOLE_0:3;
end;
