reserve a, b, c, a1, a2, b1, b2 for Int-Location,
  l, l1, l2 for Nat,
  f, g, f1, f2 for FinSeq-Location,
  i, j for Instruction of SCM+FSA,
  X, Y for set;

theorem Th4:
  MultBy(a1,b1) = MultBy(a2,b2) implies a1 = a2 & b1 = b2
proof
  assume
A1: MultBy(a1,b1) = MultBy(a2,b2);
  consider A1, B1 being Data-Location such that
A2: a1 = A1 & b1 = B1 & MultBy(a1,b1) = MultBy(A1,B1) by SCMFSA_2:def 11;
  consider A2, B2 being Data-Location such that
A3: a2 = A2 & b2 = B2 & MultBy(a2,b2) = MultBy(A2,B2) by SCMFSA_2:def 11;
A4: <*A2,B2*>.1 = A2 & <*A2,B2*>.2 = B2;
  <*A1,B1*>.1 = A1 & <*A1,B1*>.2 = B1;
  hence thesis by A1,A2,A3,A4,XTUPLE_0:3;
end;
