
theorem
  for C1,C2 being Coherence_Space holds union LinCoh(C1,C2) = [:union C1
  , union C2:]
proof
  let C1,C2 be Coherence_Space;
  thus union LinCoh(C1,C2) c= [:union C1, union C2:]
  proof
    let x be object;
    assume x in union LinCoh(C1,C2);
    then consider a being set such that
A1: x in a and
A2: a in LinCoh(C1,C2) by TARSKI:def 4;
    ex f being U-linear Function of C1,C2 st a = LinTrace f by A2,Def20;
    hence thesis by A1;
  end;
  let x,y be object;
  assume
A3: [x,y] in [:union C1, union C2:];
  then
A4: y in union C2 by ZFMISC_1:87;
  x in union C1 by A3,ZFMISC_1:87;
  then ex f being U-linear Function of C1,C2 st LinTrace f = {[x,y]} by A4,Th59
;
  then [x,y] in {[x,y]} & {[x,y]} in LinCoh(C1,C2) by Def20,TARSKI:def 1;
  hence thesis by TARSKI:def 4;
end;
