
theorem Th36:
  for C1, C2 being Coherence_Space for f,g being U-stable Function
  of C1,C2 st Trace f c= Trace g for a being Element of C1 holds f.a c= g.a
proof
  let C1, C2 be Coherence_Space;
  let f,g be U-stable Function of C1,C2;
  assume
A1: Trace f c= Trace g;
  let x be Element of C1;
A2: dom f = C1 by FUNCT_2:def 1;
  let z be object;
  assume z in f.x;
  then consider b being set such that
  b is finite and
A3: b c= x and
A4: z in f.b and
A5: for c being set st c c= x & z in f.c holds b c= c by A2,Th22;
  reconsider b as Element of C1 by A3,CLASSES1:def 1;
  now
    let c be set;
    assume that
    c in dom f and
A6: c c= b and
A7: z in f.c;
    c c= x by A3,A6;
    then b c= c by A5,A7;
    hence b = c by A6;
  end;
  then [b,z] in Trace f by A2,A4,Th31;
  then
A8: z in g.b by A1,Th31;
  dom g = C1 by FUNCT_2:def 1;
  then g.b c= g.x by A3,Def11;
  hence thesis by A8;
end;
