reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem Th11:
  x in dom(g*f) iff x in dom f & f.x in dom g
proof
  set h = g*f;
  hereby
    assume x in dom h;
    then consider y being object such that
A1: [x,y] in h by XTUPLE_0:def 12;
    consider z being object such that
A2: [x,z] in f and
A3: [z,y] in g by A1,RELAT_1:def 8;
    reconsider z as set by TARSKI:1;
    thus x in dom f by A2,XTUPLE_0:def 12;
    then z = f.x by A2,Def2;
    hence f.x in dom g by A3,XTUPLE_0:def 12;
  end;
  assume
A4: x in dom f;
  then consider z being object such that
A5: [x,z] in f by XTUPLE_0:def 12;
  assume f.x in dom g;
  then consider y being object such that
A6: [f.x,y] in g by XTUPLE_0:def 12;
    reconsider z as set by TARSKI:1;
  z = f.x by A4,A5,Def2;
  then [x,y] in h by A5,A6,RELAT_1:def 8;
  hence thesis by XTUPLE_0:def 12;
end;
