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 Th29:
  for PA,PB,PC being a_partition of Y st
  PA '<' PC & PB '<' PC holds PA '\/' PB '<' PC
proof
  let PA,PB,PC be a_partition of Y;
  assume PA '<' PC & PB '<' PC; then
A1: ERl(PA) c= ERl(PC) & ERl(PB) c= ERl(PC) by Th20;
A2: ERl(PA '\/' PB) = ERl(PA) "\/" ERl(PB) by Th23;
  ERl(PA) \/ ERl(PB) c= ERl(PC) by A1,XBOOLE_1:8;
  then ERl(PA) "\/" ERl(PB) c= ERl(PC) by EQREL_1:def 2;
  hence thesis by A2,Th20;
end;
