reserve a,b,c,d,e,f,g,h for set;

theorem
  for c1,x1,x2,x3,x4,y1,y2,y3,y4,c2,c3,c4,c5,n1,n2,n3,n4,n,c5b being set
  holds (MAJ3(x1,y1,c1) is non empty implies c2 is non empty) & (MAJ3(x2,y2,c2)
is non empty implies c3 is non empty) & (MAJ3(x3,y3,c3) is non empty implies c4
is non empty) & (MAJ3(x4,y4,c4) is non empty implies c5 is non empty) & (n1 is
non empty implies OR2(x1,y1) is non empty)& (n2 is non empty implies OR2(x2,y2)
  is non empty)& (n3 is non empty implies OR2(x3,y3) is non empty)& (n4 is non
empty implies OR2(x4,y4) is non empty)& (n is non empty implies AND5(c1,n1,n2,
n3,n4) is non empty)& (c5b is non empty iff OR2(c5,n) is non empty) implies (c5
  is non empty iff c5b is non empty)
proof
  let c1,x1,x2,x3,x4,y1,y2,y3,y4,c2,c3,c4,c5,n1,n2,n3,n4,n,c5b be set;
  thus (MAJ3(x1,y1,c1) is non empty implies c2 is non empty) & (MAJ3(x2,y2,c2)
is non empty implies c3 is non empty) & (MAJ3(x3,y3,c3) is non empty implies c4
is non empty) & (MAJ3(x4,y4,c4) is non empty implies c5 is non empty) & (n1 is
non empty implies OR2(x1,y1) is non empty)& (n2 is non empty implies OR2(x2,y2)
  is non empty)& (n3 is non empty implies OR2(x3,y3) is non empty)& (n4 is non
empty implies OR2(x4,y4) is non empty)& (n is non empty implies AND5(c1,n1,n2,
n3,n4) is non empty)& (c5b is non empty iff OR2(c5,n) is non empty) implies (c5
  is non empty iff c5b is non empty)
proof
  assume that
A1: ( MAJ3(x1,y1,c1) is non empty implies c2 is non empty)&( MAJ3(x2,y2,
  c2) is non empty implies c3 is non empty) &( MAJ3(x3,y3,c3) is non empty
  implies c4 is non empty) and
A2: MAJ3(x4,y4,c4) is non empty implies c5 is non empty and
A3: n1 is non empty implies OR2(x1,y1) is non empty and
A4: n2 is non empty implies OR2(x2,y2) is non empty and
A5: n3 is non empty implies OR2(x3,y3) is non empty and
A6: n4 is non empty implies OR2(x4,y4) is non empty and
A7: n is non empty implies AND5(c1,n1,n2,n3,n4) is non empty and
A8: c5b is non empty implies OR2(c5,n) is non empty and
A9: OR2(c5,n) is non empty implies c5b is non empty;
A10: now
    assume
A11: n is non empty;
    then
A12: c1 is non empty by A7,Def18;
A13: x3 is non empty or y3 is non empty by A5,A7,A11,Def18;
A14: x2 is non empty or y2 is non empty by A4,A7,A11,Def18;
A15: x4 is non empty or y4 is non empty by A6,A7,A11,Def18;
    x1 is non empty or y1 is non empty by A3,A7,A11,Def18;
    hence MAJ3(x4,y4,c4) is non empty by A1,A12,A14,A13,A15;
  end;
  thus c5 is non empty implies c5b is non empty by A9;
  assume c5b is non empty;
  hence thesis by A2,A8,A10;
end;
end;
