reserve
  I for non empty set,
  F for associative Group-like multMagma-Family of I,
  i, j for Element of I;

theorem
  for x,y be Element of product F st i <> j &
  x in ProjGroup(F,i) & y in ProjGroup(F,j) holds x*y = y*x
  proof
    set G = product F;
    let x,y be Element of G;
    assume A1: i <> j & x in ProjGroup(F,i) & y in ProjGroup(F,j);
    A2: the carrier of ProjGroup(F,i) = ProjSet(F,i)
    & the carrier of ProjGroup(F,j) = ProjSet(F,j) by Def2;
    A3: x in ProjSet(F,i) & y in ProjSet(F,j) by A2,A1,STRUCT_0:def 5;
    consider xx be Function,gx be Element of (F.i) such that
    A4: xx=x & dom xx = I & xx.i = gx &
    for k be Element of I st k <> i holds xx.k = 1_F.k by A3,Th2;
    consider yy be Function,gy be Element of (F.j) such that
    A5: yy=y & dom yy = I & yy.j = gy &
    for k be Element of I st k <> j holds yy.k = 1_F.k by A3,Th2;
    A6: the carrier of product F = product Carrier F by GROUP_7:def 2;
    set xy=x*y;
    set yx=y*x;
    A7: dom xy = I by A6,PARTFUN1:def 2;
    A8: dom yx = I by A6,PARTFUN1:def 2;
    for k be object st k in dom xy holds xy.k = yx.k
    proof
      let k0 be object;
      assume k0 in dom xy;
      then reconsider k=k0 as Element of I by A6,PARTFUN1:def 2;
      per cases;
      suppose A9: k0 <> i & k0 <> j; then
A10:    xx.k = 1_F.k by A4;
A11:    yy.k = 1_F.k by A9,A5;
        xy.k = (1_F.k) * (1_F.k) by A4,A5,A10,A11,GROUP_7:1
        .=yx.k by A4,A5,A10,A11,GROUP_7:1;
        hence xy.k0 = yx. k0;
      end;
      suppose A12: k0 = i or k0 = j;
        per cases by A12;
        suppose A13: k0 = i; then
A14:      yy.k = 1_F.k by A1,A5;
          reconsider gx as Element of F.k by A13;
          xy.k = gx * (1_F.k) by A4,A5,A14,A13,GROUP_7:1
          .=gx by GROUP_1:def 4
          .=(1_F.k)*gx by GROUP_1:def 4
          .=yx.k by A4,A5,A14,A13,GROUP_7:1;
          hence xy.k0 = yx. k0;
        end;
        suppose A15: k0 = j; then
A16:      xx.k = 1_F.k by A1,A4;
          reconsider gy as Element of F.k by A15;
          xy.k = (1_F.k)*gy by A4,A5,A16,A15,GROUP_7:1
          .=gy by GROUP_1:def 4
          .=gy*(1_F.k) by GROUP_1:def 4
          .=yx.k by A4,A5,A16,A15,GROUP_7:1;
          hence xy.k0 = yx. k0;
        end;
      end;
    end;
    hence thesis by A7,A8,FUNCT_1:2;
  end;
