reserve a,b,d,n,k,i,j,x,s for Nat;

theorem Th10:
  for j being Nat st 0 < j <= 7 ex i being Nat st
    i>0 &
    Fib(0),Fib(i) are_congruent_mod j &
    Fib(1),Fib(i+1) are_congruent_mod j
proof
  let j be Nat such that
A1: 0 < j <= 7;
  j=0 or ... or j=7 by A1;
  then per cases by A1;
  suppose j=1;
    then Fib(0),Fib(1) are_congruent_mod j &
    Fib(1),Fib(1+1) are_congruent_mod j by INT_1:13;
    hence thesis;
  end;
  suppose
A2:   j=2;
A3:   0*j+0 mod j = 0 & j*0+1 mod j = 1 by A2;
    2*1+0 mod 2 = 0 & 2*1+1 mod 2 = 1;
    then Fib(0),Fib(3) are_congruent_mod j &
    Fib(1),Fib(3+1) are_congruent_mod j
    by A2,A3,PRE_FF:1,NAT_D:64,FIB_NUM2:22,23;
    hence thesis;
  end;
  suppose
A4:   j=3;
A5:   0*j+0 mod j = 0 & j*0+1 mod j = 1 by A4,NUMBER02:16;
    3*7+0 mod j = 0 & 3*11+1 mod j = 1 by A4,NUMBER02:16;
    then Fib(0),Fib(8) are_congruent_mod j &
    Fib(1),Fib(8+1) are_congruent_mod j
    by A4,A5,PRE_FF:1,NAT_D:64,Th7;
    hence thesis;
  end;
  suppose
A6:   j=4;
A7:   0*j+0 mod j = 0 & j*0+1 mod j = 1 by A6,NUMBER02:16;
    4*2+0 mod j = 0 & 4*3+1 mod j = 1 by A6,NUMBER02:16;
    then Fib(0),Fib(6) are_congruent_mod j &
    Fib(1),Fib(6+1) are_congruent_mod j
    by A6,A7,PRE_FF:1,NAT_D:64,Th7;
    hence thesis;
  end;
  suppose
A8:   j=5;
A9:   0*j+0 mod j = 0 & j*0+1 mod j = 1 by A8,NUMBER02:16;
    5*1353+0 mod j = 0 & 5*2189+1 mod j = 1 by A8,NUMBER02:16;
    then Fib(0),Fib(20) are_congruent_mod j &
    Fib(1),Fib(20+1) are_congruent_mod j
    by A8,A9,PRE_FF:1,NAT_D:64,Th7;
    hence thesis;
  end;
  suppose
A10:  j=6;
A11:  0*j+0 mod j = 0 & j*0+1 mod j = 1 by A10,NUMBER02:16;
    6*7728+0 mod j = 0 & 6*12504+1 mod j = 1 by A10,NUMBER02:16;
    then Fib(0),Fib(24) are_congruent_mod j &
    Fib(1),Fib(24+1) are_congruent_mod j
    by A10,A11,PRE_FF:1,NAT_D:64,Th7;
    hence thesis;
  end;
  suppose
A12:  j=7;
A13:  0*j+0 mod j = 0 & j*0+1 mod j = 1 by A12,NUMBER02:16;
    7*141+0 mod j = 0 & 7*228+1 mod j = 1 by A12,NUMBER02:16;
    then Fib(0),Fib(16) are_congruent_mod j &
    Fib(1),Fib(16+1) are_congruent_mod j
    by A12,A13,PRE_FF:1,NAT_D:64,Th7;
    hence thesis;
  end;
end;
