
theorem
  for C1,C2 being Coherence_Space holds union StabCoh(C1,C2) = [:
  Sub_of_Fin C1, union C2:]
proof
  let C1,C2 be Coherence_Space;
  thus union StabCoh(C1,C2) c= [:Sub_of_Fin C1, union C2:]
  proof
    let x be object;
    assume x in union StabCoh(C1,C2);
    then consider a being set such that
A1: x in a and
A2: a in StabCoh(C1,C2) by TARSKI:def 4;
    ex f being U-stable Function of C1,C2 st a = Trace f by A2,Def18;
    then a c= [:Sub_of_Fin C1, union C2:] by Th46;
    hence thesis by A1;
  end;
  let x,y be object;
  assume
A3: [x,y] in [:Sub_of_Fin C1, union C2:];
  then
A4:  y in union C2 by ZFMISC_1:87;
   reconsider x as set by TARSKI:1;
A5: x in Sub_of_Fin C1 by A3,ZFMISC_1:87;
  then x is finite by Def3;
  then ex f being U-stable Function of C1,C2 st Trace f = {[x,y]} by A5,A4,Th42
;
  then [x,y] in {[x,y]} & {[x,y]} in StabCoh(C1,C2) by Def18,TARSKI:def 1;
  hence thesis by TARSKI:def 4;
end;
