
theorem THJC:
  for n being Nat
  for r,s being Real, u,v,w being Element of TOP-REAL n st
    r * u - r * v = s * w - s * u holds (r + s) * u = r * v + s * w
  proof
    let n be Nat;
    let r,s be Real, u,v,w be Element of TOP-REAL n;
    assume r * u - r * v = s * w - s * u;
    then r * u - r * v + s * u = s * w + (- s * u + s * u) by RVSUM_1:15
                              .= s * w + ((- 1)* s * u + s * u) by RVSUM_1:49
                              .= s * w + (- s + s) * u by RVSUM_1:50
                              .= s * w by THJE;
    then s * w + r * v = r * u + ((-1)* r * v) + s * u + r * v by RVSUM_1:49
                      .= r * u + s * u + ((-1)* r * v) + r * v by RVSUM_1:15
                      .= r * u + s * u + (((-1)* r * v) + r * v) by RVSUM_1:15
                      .= r * u + s * u + (- r + r) * v  by RVSUM_1:50
                      .= (r * u + s * u) by THJE;
    hence thesis by RVSUM_1:50;
  end;
