
theorem ::2JCWR:
  for s0,s1,s2,s3,ns0,ns1,ns2,ns3,q1,q2,nq1,nq2,R being set holds (s0 is
not empty iff AND2(NOT1 q2, NOT1 q1) is not empty)& (s1 is not empty iff AND2(
  NOT1 q2, q1) is not empty)& (s2 is not empty iff AND2( q2, NOT1 q1) is not
  empty)& (s3 is not empty iff AND2( q2, q1) is not empty) & (ns0 is not empty
iff AND2(NOT1 nq2,NOT1 nq1) is not empty)& (ns1 is not empty iff AND2(NOT1 nq2,
nq1) is not empty)& (ns2 is not empty iff AND2( nq2,NOT1 nq1) is not empty)& (
ns3 is not empty iff AND2( nq2, nq1) is not empty) & (nq1 is not empty iff AND2
  (NOT1 q2, R) is not empty)& (nq2 is not empty iff AND2( q1, R) is not empty)
implies (ns1 is not empty iff AND2(s0, R) is not empty)& (ns3 is not empty iff
  AND2(s1, R) is not empty)& (ns2 is not empty iff AND2(s3, R) is not empty)& (
  ns0 is not empty iff OR2(AND2(s2, R),NOT1 R) is not empty)
proof
  let s0,s1,s2,s3,ns0,ns1,ns2,ns3,q1,q2,nq1,nq2,R be set;
  assume that
A1: s0 is not empty iff AND2(NOT1 q2,NOT1 q1) is not empty and
A2: s1 is not empty iff AND2(NOT1 q2,q1) is not empty and
A3: s2 is not empty iff AND2(q2,NOT1 q1) is not empty and
A4: s3 is not empty iff AND2(q2,q1) is not empty and
A5: ns0 is not empty iff AND2(NOT1 nq2,NOT1 nq1) is not empty and
A6: ns1 is not empty iff AND2(NOT1 nq2,nq1) is not empty and
A7: ns2 is not empty iff AND2(nq2,NOT1 nq1) is not empty and
A8: ns3 is not empty iff AND2(nq2,nq1) is not empty and
A9: ( nq1 is not empty iff AND2(NOT1 q2,R) is not empty)&( nq2 is not
  empty iff AND2( q1,R) is not empty);
  ns1 is not empty iff NOT1 q2 is not empty & R is not empty & not(q1 is
  not empty & R is not empty) by A6,A9;
  hence ns1 is not empty iff AND2(s0,R) is not empty by A1;
  ns3 is not empty iff NOT1 q2 is not empty & q1 is not empty & R is not
  empty by A8,A9;
  hence ns3 is not empty iff AND2(s1,R) is not empty by A2;
  ns2 is not empty iff not(NOT1 q2 is not empty & R is not empty) & q1 is
  not empty & R is not empty by A7,A9;
  then
  ns2 is not empty iff q2 is not empty & q1 is not empty & R is not empty;
  hence ns2 is not empty iff AND2(s3,R) is not empty by A4;
  ns0 is not empty iff not(NOT1 q2 is not empty & R is not empty) & not(
  q1 is not empty & R is not empty) by A5,A9;
  then ns0 is not empty iff q2 is not empty & NOT1 q1 is not empty & R is not
  empty or not R is not empty;
  hence thesis by A3;
end;
