theorem Th48:
  p1 is_an_universal_closure_of p & q1 is_an_universal_closure_of
  q implies (p |- q iff 'not' q1 |- 'not' p1)
proof
  assume that
A1: p1 is_an_universal_closure_of p and
A2: q1 is_an_universal_closure_of q;
  now
    q |-| q1 by A2,Th38;
    then
A3: q |- q1;
    p1 |-| p by A1,Th38;
    then
A4: p1 |- p;
    assume p |- q;
    then p1 |- q by A4,Th6;
    then
A5: p1 |- q1 by A3,Th6;
    p1 is closed by A1;
    hence 'not' q1 |- 'not' p1 by A5,Th43;
  end;
  hence p |- q implies 'not' q1 |- 'not' p1;
  now
    q1 |-| q by A2,Th38;
    then
A6: q1 |- q;
    p1 |-| p by A1,Th38;
    then
A7: p |- p1;
    assume
A8: 'not' q1 |- 'not' p1;
    q1 is closed by A2;
    then p1 |- q1 by A8,Th45;
    then p |- q1 by A7,Th6;
    hence p |- q by A6,Th6;
  end;
  hence thesis;
end;
