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 Th12:
  x in dom(g*f) implies (g*f).x = g.(f.x)
proof
  set h = g*f;
  assume
A1: x in dom h;
  then consider y being object such that
A2: [x,y] in h by XTUPLE_0:def 12;
  consider z being object such that
A3: [x,z] in f and
A4: [z,y] in g by A2,RELAT_1:def 8;
    reconsider z,y as set by TARSKI:1;
  x in dom f by A3,XTUPLE_0:def 12;
  then
A5: z = f.x by A3,Def2;
  then f.x in dom g by A4,XTUPLE_0:def 12;
  then y = g.(f.x) by A4,A5,Def2;
  hence thesis by A1,A2,Def2;
end;
