reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem Th46:
  a>=1 & (a+1)|^2 + ((a+1)+x)|^2 <= ((a+1)+x+1)|^2 implies
  a|^2 + (a+x)|^2 < (a+x+1)|^2
  proof
    A2: (a+x+1)|^2 = a|^2+x|^2+1|^2 + 2*a*x + 2*a*1 + 2*x*1 by SERIES_4:1
    .= a|^2 + x|^2 + 2*a*x + 2*x + 2*a + 1;
    A3: (a+x+2)|^2 = a|^2+x|^2 + 2|^2 + 2*a*x + 2*a*2 + 2*x*2 by SERIES_4:1
    .= 2*a*2 + 2*x + 2*x + 2|^2 + a|^2 + 2*a*x + x|^2;
    A4: (a+x)|^2 = a|^2 + 2*a*x + x|^2 by Lm10;
    assume
    A5: a>=1 & (a+1)|^2 + ((a+1)+x)|^2 <= ((a+1)+x+1)|^2; then
    A5a: (a+1)|^2 + (a+x+1)|^2 -(a+x+2)|^2 <=
    (a+x+2)|^2 -(a+x+2)|^2 by XREAL_1:9;
    -((a+1)|^2 + (a+x+1)|^2 -(a+x+2)|^2) =
       (a+x+2)|^2 - ((a+1)|^2 + (a+x+1)|^2)
    .= a|^2 + x|^2 + 2|^2 + 2*a*x + 2*2*a + 2*x + 2*x -
    ((a|^2 + 2*a + 1) + (a|^2 + x|^2 + 1|^2 + 2*a*x + 2*a*1 + 2*x*1))
    by Lm10,A2,A3
    .= 2|^2 + 2*x - 1 - a|^2 - 1*1
    .= 2*2 + 2*x - 1 -a|^2 - 1 by WSIERP_1:1; then
    A6: (a|^2 + (a+x)|^2) + 0 <=
    (a|^2 + (a+x)|^2) + (4 + 2*x - a|^2 - 2) by A5a,XREAL_1:6;
    A7: a + 1 > 0 + 1 & a + 1 >= 1 + 1 by A5, XREAL_1:6; then
    a + (a + 1) > a + (1 + 0) by XREAL_1:6; then
    2*a + 1 > 1 + 1 by A7,XXREAL_0:2; then
    (a|^2 + x|^2 + 2*a*x + 2*x) + (2*a + 1) >
      (a|^2 + x|^2 + 2*a*x + 2*x) + 2 by XREAL_1:6;
    hence thesis by A2,A4,A6,XXREAL_0:2;
  end;
