
theorem Th46:
  for C1,C2 being Coherence_Space, f being U-stable Function of C1
  ,C2 holds Trace f c= [:Sub_of_Fin C1, union C2:]
proof
  let C1,C2 be Coherence_Space, f be U-stable Function of C1,C2;
  let x1,x2 be object;
  assume
A1: [x1,x2] in Trace f;
  then consider a, y being set such that
A2: [x1,x2] = [a,y] and
A3: a in dom f and
A4: y in f.a and
  for b being set st b in dom f & b c= a & y in f.b holds a = b by Def17;
A5: dom f = C1 by FUNCT_2:def 1;
  then a is finite by A1,A2,Th33;
  then
A6: a in Sub_of_Fin C1 by A3,A5,Def3;
  y in union C2 by A3,A4,A5,Lm1;
  hence thesis by A2,A6,ZFMISC_1:87;
end;
