reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th105:
  0 <= j & j^2 < z < (j+1)^2 implies not ex i being Integer st z = i^2
  proof
    assume that
A1: j >= 0 and
A2: j^2 < z and
A3: z < (j+1)^2;
    given i such that
A4: i^2 = z;
    sqrt (j^2) = j by A1,SQUARE_1:22;
    then
A5: j < sqrt z by A2,SQUARE_1:27;
    sqrt (j+1)^2 = j+1 by A1,SQUARE_1:22;
    then sqrt(i^2) < j+1 by A3,A4,SQUARE_1:27;
    hence thesis by A4,A5,INT_1:7;
  end;
