reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;

theorem Th29:
  i <= 0 implies i * h = -( |.i.| * h)
proof
  assume
A1: i <= 0;
  per cases by A1;
  suppose
    i < 0;
    hence thesis by Def8;
  end;
  suppose
A2: i = 0;
    hence i * h = 0_G by Lm3
      .= -(0_G) by Th8
      .= -( 0 * h) by Def7
      .= -( |.i.| * h) by A2,ABSVALUE:def 1;
  end;
end;
