
theorem Th1:
  for X,x being set holds x in FlatCoh X iff x = {} or ex y being
  set st x = {y} & y in X
proof
  let X,x be set;
  hereby
    assume
A1: x in FlatCoh X;
    assume x <> {};
    then reconsider y = x as non empty set;
    set z = the Element of y;
    reconsider z as set;
    take z;
    thus x = {z}
    proof
      hereby
        let c be object;
        assume c in x;
        then [z,c] in id X by A1,COH_SP:def 3;
        then c = z by RELAT_1:def 10;
        hence c in {z} by TARSKI:def 1;
      end;
      thus thesis by ZFMISC_1:31;
    end;
    [z,z] in id X by A1,COH_SP:def 3;
    hence z in X by RELAT_1:def 10;
  end;
A2: now
    given a being set such that
A3: x = {a} and
A4: a in X;
    let y,z be set;
    assume y in x & z in x;
    then y = a & z = a by A3,TARSKI:def 1;
    hence [y,z] in id X by A4,RELAT_1:def 10;
  end;
  assume x = {} or ex y being set st x = {y} & y in X;
  hence thesis by A2,COH_SP:1,def 3;
end;
