reserve A for non trivial Nat,
        B,C,n,m,k for Nat,
        e for Nat;

theorem Th6:
  for T1,T2 be _Theta,epsilon1,epsilon2 be Real st
    0 <= epsilon1 & 0 <= epsilon2
  holds ex T be _Theta st T1*epsilon1 + T2*epsilon2 = T* (epsilon1+epsilon2)
proof
  let T1,T2 be _Theta,epsilon1,epsilon2 be Real such that
A1: 0 <= epsilon1 & 0 <= epsilon2;
  reconsider I=1 as _Theta by Def1;
  -1<= T1 <= 1 & -1 <=T2<=1 by Def1;
  then (-1)*epsilon1 <= T1*epsilon1 <= 1*epsilon1 &
  (-1)*epsilon2 <= T2*epsilon2 <= 1*epsilon2 by A1,XREAL_1:64;
  then (-1)*epsilon1 + (-1)*epsilon2 <= T1*epsilon1 + T2*epsilon2
    <= 1*epsilon1 + 1*epsilon2 by XREAL_1:7;
  then (-I)*(epsilon1 + epsilon2) <= T1*epsilon1 + T2*epsilon2
    <= I*(epsilon1+epsilon2);
  hence thesis by Th4;
end;
