 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
reserve G1,G2 for Group;

theorem ThCommutationEq: :: TH95
  for G being strict commutative Group
  for a,b being Element of INT.Group 2 st b = 0
  holds ((inversions G).b) * ((inversions G).a) = (inversions G).a
  & ((inversions G).a) * ((inversions G).b) = (inversions G).a
proof
  let G be strict commutative Group;
  let a,b be Element of INT.Group 2;
  assume b = 0;
  then A1: ((inversions G).b) = id G by DefInversions;
  a in INT.Group 2;
  then per cases by EltsOfINTGroup2;
  suppose a = 0;
    then A2: ((inversions G).a) = id G by DefInversions;
    thus ((inversions G).b) * ((inversions G).a)
     = (id G) * (id G) by A1,A2,AUTGROUP:8
    .= (inversions G).a by A2, FUNCT_2:17;
    thus ((inversions G).a) * ((inversions G).b)
     = (id G) * (id G) by A1,A2,AUTGROUP:8
    .= (inversions G).a by A2, FUNCT_2:17;
  end;
  suppose a = 1;
    then A3: ((inversions G).a) = inverse_op G by DefInversions;
    hence ((inversions G).b) * ((inversions G).a)
     = (id G) * (inverse_op G) by A1,AUTGROUP:8
    .= (inversions G).a by A3, FUNCT_2:17;
    thus ((inversions G).a) * ((inversions G).b)
     = (inverse_op G) * (id G) by A1,A3,AUTGROUP:8
    .= (inversions G).a by A3, FUNCT_2:17;
  end;
end;
