reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th90:
  for k be positive Nat holds t mod k = k - 1 iff (t+1) mod k = 0
  proof
    let k be positive Nat;
    thus t mod k = k - 1 implies (t+1) mod k = 0
    proof
      assume
      A1: t mod k = k - 1;
      0 = (0 + 1*k) mod k
      .= (1 + (t mod k)) mod k by A1
      .= (1 + t) mod k by Th88;
      hence thesis;
    end;
    assume
    A1: (t+1) mod k = 0;
    (k-1)+1 > (k-1)+0 by XREAL_1:6; then
    k - 1 = ((k -1) + ((t+1) mod k)) mod k by A1,NAT_D:24
      .= ((k -1) + (t +1)) mod k by Th88
      .= (t + 1*k) mod k
      .= t mod k by NAT_D:61;
    hence thesis;
  end;
