theorem
  R1 + R = R2 + R implies R1 = R2
proof
  assume R1 + R = R2 + R;
  then R1 + (R + -R)= (R2 + R)+-R by FINSEQOP:28;
  then
A1: R1 + (R + -R)= R2 + (R + -R) by FINSEQOP:28;
  R + -R = i|->0 by Th8,Th9,BINOP_2:2,FINSEQOP:73;
  then R1 = R2 + (i|->(0 qua Real)) by A1,BINOP_2:2,FINSEQOP:56;
  hence thesis by BINOP_2:2,FINSEQOP:56;
end;
