reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;

theorem Th1:
  x in R-Seg a iff x <> a & [x,a] in R
proof
  hereby
    assume
A1: x in R-Seg a;
    hence x <> a by ZFMISC_1:56;
    ex y being object st [x,y] in R & y in {a} by A1,RELAT_1:def 14;
    hence [x,a] in R by TARSKI:def 1;
  end;
  assume that
A2: x <> a and
A3: [x,a] in R;
  a in {a} by TARSKI:def 1;
  then x in Coim(R,a) by A3,RELAT_1:def 14;
  hence thesis by A2,ZFMISC_1:56;
end;
