
theorem
  for x0,x1,y0,y1,z0,z1,z2,z3,q00,q01,c01,q11,c11 being set holds (q00
is not empty iff AND2(x0,y0) is not empty)& (q01 is not empty iff XOR3(AND2(x1,
y0),AND2(x0,y1),{} ) is not empty)& (c01 is not empty iff MAJ3(AND2(x1,y0),AND2
(x0,y1),{} ) is not empty)& (q11 is not empty iff XOR3(AND2(x1,y1),{} ,c01) is
not empty)& (c11 is not empty iff MAJ3(AND2(x1,y1),{} ,c01) is not empty)& (z0
is not empty iff q00 is not empty)& (z1 is not empty iff q01 is not empty)& (z2
  is not empty iff q11 is not empty)& (z3 is not empty iff c11 is not empty)
  implies (z0 is not empty iff MULT210(x1,y1,x0,y0) is not empty)& (z1 is not
empty iff MULT211(x1,y1,x0,y0) is not empty)& (z2 is not empty iff MULT212(x1,
y1,x0,y0) is not empty)& (z3 is not empty iff MULT213(x1,y1,x0,y0) is not empty
  )
proof
  let x0,x1,y0,y1,z0,z1,z2,z3,q00,q01,c01,q11,c11 be set;
  assume that
A1: q00 is not empty iff AND2(x0,y0) is not empty and
A2: q01 is not empty iff XOR3(AND2(x1,y0),AND2(x0,y1),{} ) is not empty and
A3: c01 is not empty iff MAJ3(AND2(x1,y0),AND2(x0,y1),{} ) is not empty and
A4: q11 is not empty iff XOR3(AND2(x1,y1),{} ,c01) is not empty and
A5: c11 is not empty iff MAJ3(AND2(x1,y1),{} ,c01) is not empty and
A6: z0 is not empty iff q00 is not empty and
A7: z1 is not empty iff q01 is not empty and
A8: z2 is not empty iff q11 is not empty and
A9: z3 is not empty iff c11 is not empty;
  thus z0 is not empty iff MULT210(x1,y1,x0,y0) is not empty by A1,A6;
  thus z1 is not empty iff MULT211(x1,y1,x0,y0) is not empty by A2,A7;
  set m212 = MULT212(x1,y1,x0,y0);
  set x1y1 = AND2(x1,y1);
  set x0y1 = AND2(x0,y1);
  set x1y0 = AND2(x1,y0);
  m212 = XOR3({},x1y1,MAJ3(x1y0,x0y1,{})) by GATE_1:def 35;
  then
  m212 is not empty iff x1y1 is not empty & not MAJ3(x1y0,x0y1,{}) is not
  empty or not x1y1 is not empty & MAJ3(x1y0,x0y1,{}) is not empty;
  hence z2 is not empty iff MULT212(x1,y1,x0,y0) is not empty by A3,A4,A8;
  set m213 = MULT213(x1,y1,x0,y0);
  m213 = MAJ3({},x1y1,MAJ3(x1y0,x0y1,{})) by GATE_1:def 36;
  then m213 is not empty iff x1y1 is not empty & MAJ3(x1y0,x0y1,{}) is not
  empty;
  hence thesis by A3,A5,A9;
end;
