reserve C1, C2 for Coherence_Space;

theorem Th96:
  for C1,C2 being Coherence_Space, x being set holds x in C1 [*]
  C2 iff ex a being Element of C1, b being Element of C2 st x c= [:a,b:]
proof
  let C1,C2 be Coherence_Space, x be set;
  hereby
    assume x in C1[*]C2;
    then consider y being set such that
A1: x in y and
A2: y in the set of all
bool [:a,b:] where a is Element of C1, b is Element of C2 by TARSKI:def 4;
    consider a being Element of C1, b being Element of C2 such that
A3: y = bool [:a,b:] by A2;
    take a,b;
    thus x c= [:a,b:] by A1,A3;
  end;
  given a9 being Element of C1, b9 being Element of C2 such that
A4: x c= [:a9,b9:];
  bool [:a9,b9:] in the set of all
bool [:a,b:] where a is Element of C1, b is Element
  of C2;
  hence thesis by A4,TARSKI:def 4;
end;
