
theorem Th55:
  for C1, C2 being Coherence_Space for f,g being U-linear Function
  of C1,C2 st LinTrace f = LinTrace g holds f = g
proof
  let C1, C2 be Coherence_Space;
  let f,g be U-linear Function of C1,C2;
  assume
A1: LinTrace f = LinTrace g;
  Trace f = Trace g
  proof
    let a,y be object;
     reconsider aa=a as set by TARSKI:1;
    hereby
      assume
A2:   [a,y] in Trace f;
      then consider x being set such that
A3:   aa = {x} by Th49;
      [x,y] in LinTrace f by A2,A3,Th50;
      hence [a,y] in Trace g by A1,A3,Th50;
    end;
    assume
A4: [a,y] in Trace g;
    then consider x being set such that
A5: aa = {x} by Th49;
    [x,y] in LinTrace g by A4,A5,Th50;
    hence thesis by A1,A5,Th50;
  end;
  hence thesis by Th37;
end;
