 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 ThCommutationEq2: :: TH96
  for G being strict commutative Group
  for a,b being Element of INT.Group 2 st a = 1 & b = 1
  holds ((inversions G).b) * ((inversions G).a) = (inversions G).(a * b)
proof
  let G be strict commutative Group;
  let a,b be Element of INT.Group 2;
  assume A1: a = 1;
  assume A2: b = 1; then
  A3: a * b = 1_(INT.Group 2) by A1,ThMultTableINTGroup2
           .= 0 by GR_CY_1:14;
  ((inversions G).b) = inverse_op G & ((inversions G).a) = inverse_op G
    by A1,A2,DefInversions;
  hence ((inversions G).b) * ((inversions G).a)
   = (inverse_op G) * (inverse_op G) by AUTGROUP:8
  .= id G by ThInverseOpGSquaresToIdentity
  .= (inversions G).(a * b) by A3, DefInversions;
end;
