reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

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;
