
theorem Th3:
  for A being non empty set st A is c=-linear holds [:union A,
  union A:] = union { [:a,a:] where a is Element of A : a in A }
proof
  let A be non empty set;
  set X = { [:a,a:] where a is Element of A : a in A }, Y = { [:a,b:] where a
  is Element of A, b is Element of A : a in A & b in A };
  assume
A1: A is c=-linear;
A2: union Y c= union X
  proof
    let Z be object;
    assume Z in union Y;
    then consider z being set such that
A3: Z in z and
A4: z in Y by TARSKI:def 4;
    consider a,b being Element of A such that
A5: z = [:a,b:] and
    a in A and
    b in A by A4;
A6: a,b are_c=-comparable by A1;
    per cases by A6;
    suppose
A7:   a c= b;
A8:   [:b,b:] in X;
      [:a,b:] c= [:b,b:] by A7,ZFMISC_1:95;
      hence thesis by A3,A5,A8,TARSKI:def 4;
    end;
    suppose
A9:   b c= a;
A10:  [:a,a:] in X;
      [:a,b:] c= [:a,a:] by A9,ZFMISC_1:95;
      hence thesis by A3,A5,A10,TARSKI:def 4;
    end;
  end;
  X c= Y
  proof
    let Z be object;
    assume Z in X;
    then ex a being Element of A st Z = [:a,a:] & a in A;
    hence thesis;
  end;
  then union X c= union Y by ZFMISC_1:77;
  then union X = union Y by A2;
  hence thesis by Th2;
end;
