reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;

theorem Th6:
  proj1 {[x,y]} = {x} & proj2 {[x,y]} = {y}
proof
  thus proj1 {[x,y]} c= {x}
  proof
    let z be object;
    assume z in proj1 {[x,y]};
    then consider t being object such that
A1: [z,t] in {[x,y]} by XTUPLE_0:def 12;
    [z,t] = [x,y] by A1,TARSKI:def 1;
    then z = x by XTUPLE_0:1;
    hence thesis by TARSKI:def 1;
  end;
  thus {x} c= proj1 {[x,y]}
  proof
    let z be object;
    assume z in {x};
    then z = x by TARSKI:def 1;
    then [z,y] in {[x,y]} by TARSKI:def 1;
    hence thesis by XTUPLE_0:def 12;
  end;
  thus proj2 {[x,y]} c= {y}
  proof
    let z be object;
    assume z in proj2 {[x,y]};
    then consider t being object such that
A2: [t,z] in {[x,y]} by XTUPLE_0:def 13;
    [t,z] = [x,y] by A2,TARSKI:def 1;
    then z = y by XTUPLE_0:1;
    hence thesis by TARSKI:def 1;
  end;
  thus {y} c= proj2 {[x,y]}
  proof
    let z be object;
    assume z in {y};
    then z = y by TARSKI:def 1;
    then [x,z] in {[x,y]} by TARSKI:def 1;
    hence thesis by XTUPLE_0:def 13;
  end;
end;
