reserve AS for AffinSpace,
  a,b,c,d,p,q,r,s,x for Element of AS;
reserve AFP for AffinPlane,
  a,a9,b,b9,c,c9,d,p,p9,q,q9,r,x,x9,y,y9,z for Element of AFP,
  A,C,P for Subset of AFP,
  f,g,h,f1,f2 for Permutation of the carrier of AFP;

theorem Th10:
  AFP is translational & f is translation & g is translation implies f*g=g*f
proof
  assume that
A1: AFP is translational and
A2: f is translation and
A3: g is translation;
A4: g is dilatation by A3,TRANSGEO:def 11;
  now
    set a = the Element of AFP;
    assume that
A5: f<>id the carrier of AFP and
A6: g<>id the carrier of AFP;
A7: a<>f.a by A2,A5,TRANSGEO:def 11;
A8: a<>g.a by A3,A6,TRANSGEO:def 11;
    now
      consider b such that
A9:   not LIN a,f.a,b by A7,AFF_1:13;
      consider h such that
A10:  h is translation and
A11:  h.a=b by A1,Th7;
A12:  h *g is translation by A3,A10,TRANSGEO:86;
      assume
A13:  LIN a,f.a,g.a;
      not LIN a,f.a,h.(g.a)
      proof
A14:    not LIN a,g.a,b
        proof
          assume
A15:      LIN a,g.a,b;
          LIN a,g.a,f.a & LIN a,g.a,a by A13,AFF_1:6,7;
          hence contradiction by A8,A9,A15,AFF_1:8;
        end;
        then (g*h).a=(h*g).a by A3,A10,A11,Th9;
        then (g*h).a=h.(g.a) by FUNCT_2:15;
        then
A16:    g.b=h.(g.a) by A11,FUNCT_2:15;
        assume
A17:    LIN a,f.a,h.(g.a);
        a,g.a // b,g.b & a,b // g.a,g.b by A3,A4,TRANSGEO:68,82;
        then LIN a,f.a,a & not LIN g.a,a,h.(g.a) by A14,A16,Lm5,AFF_1:7;
        hence contradiction by A7,A13,A17,AFF_1:8;
      end;
      then
A18:  not LIN a,f.a,(h*g).a by FUNCT_2:15;
      h*(f*g)=(h*f)*g by RELAT_1:36
        .=(f*h)*g by A2,A9,A10,A11,Th9
        .=f*(h*g) by RELAT_1:36
        .=(h*g)*f by A2,A12,A18,Th9
        .=h*(g*f) by RELAT_1:36;
      hence thesis by TRANSGEO:5;
    end;
    hence thesis by A2,A3,Th9;
  end;
  hence thesis by TRANSGEO:4;
end;
