reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th91:
  for a,b,a1,a2,a3,b1,b2,b3 being Real st
  a > 0 & b > 0 & a1 >= 1 & a2 > 0 & a3 >= 0 & b1 > 0 & b2 >= 1 & b3 >= 0 holds
  for m,n being Nat st m > n holds
  (recSeqCart(a,b,a1,a2,a3,b1,b2,b3).m)`1
  > (recSeqCart(a,b,a1,a2,a3,b1,b2,b3).n)`1 &
  (recSeqCart(a,b,a1,a2,a3,b1,b2,b3).m)`2
  > (recSeqCart(a,b,a1,a2,a3,b1,b2,b3).n)`2
  proof
    let a,b,a1,a2,a3,b1,b2,b3 be Real such that
A1: a > 0 & b > 0 and
A2: a1 >= 1 & a2 > 0 & a3 >= 0 and
A3: b1 > 0 & b2 >= 1 & b3 >= 0;
    set f = recSeqCart(a,b,a1,a2,a3,b1,b2,b3);
    let m,n such that
A4: m > n;
    defpred P[Nat] means
    $1 > n implies (f.$1)`1 > (f.n)`1 & (f.$1)`2 > (f.n)`2;
A5: P[0];
A6: P[k] implies P[k+1]
    proof
      assume that
A7:   P[k] and
A8:   k+1 > n;
A9:   f.(k+1) = [a1*(f.k)`1+a2*(f.k)`2+a3,b1*(f.k)`1+b2*(f.k)`2+b3] by Def10;
      (f.k)`1 >= 0 by A1,A2,A3,Th90;
      then
A10:  1*(f.k)`1 <= a1*(f.k)`1 by A2,XREAL_1:64;
      (f.k)`2 > 0 by A1,A2,A3,Th89;
      then
B11:  a1*(f.k)`1+0 < a1*(f.k)`1+a2*(f.k)`2 by A2,XREAL_1:6;
      a1*(f.k)`1+a2*(f.k)`2+0 <= a1*(f.k)`1+a2*(f.k)`2+a3 by A2,XREAL_1:6;
      then
A11:  a1*(f.k)`1+0 < a1*(f.k)`1+a2*(f.k)`2+a3 by B11,XXREAL_0:2;
      (f.k)`2 >= 0 by A1,A2,A3,Th90;
      then
A12:  1*(f.k)`2 <= b2*(f.k)`2 by A3,XREAL_1:64;
      (f.k)`1 > 0 by A1,A2,A3,Th89;
      then
B13:  b2*(f.k)`2+0 < b1*(f.k)`1+b2*(f.k)`2 by A3,XREAL_1:6;
      b1*(f.k)`1+b2*(f.k)`2+0 <= b1*(f.k)`1+b2*(f.k)`2+b3 by A3,XREAL_1:6;
      then
A13:  b2*(f.k)`2+0 < b1*(f.k)`1+b2*(f.k)`2+b3 by B13,XXREAL_0:2;
      k >= n by A8,NAT_1:13;
      then per cases by XXREAL_0:1;
      suppose
A14:    k > n;
        then (f.n)`1 < a1*(f.k)`1 by A7,A10,XXREAL_0:2;
        hence (f.(k+1))`1 > (f.n)`1 by A9,A11,XXREAL_0:2;
        (f.n)`2 < b2*(f.k)`2 by A7,A12,A14,XXREAL_0:2;
        hence (f.(k+1))`2 > (f.n)`2 by A9,A13,XXREAL_0:2;
      end;
      suppose k = n;
        hence thesis by A9,A10,A11,A12,A13,XXREAL_0:2;
      end;
    end;
    P[k] from NAT_1:sch 2(A5,A6);
    hence thesis by A4;
  end;
