 reserve i,j,k,m,n,m1,n1 for Nat;
 reserve a,r,r1,r2 for Real;
 reserve m0,cn,cd for Integer;
 reserve x1,x2,o for object;
 reserve t for 1_greater Nat;

theorem
  ex x,y be Integer st |. x*a - y .| < 1/t & 0 < x & x <= t
proof
    consider x1, x2 be object such that
A1:  x1 in dom (F_dp1(t,a)) and
A2:  x2 in dom (F_dp1(t,a)) and
A3:  x1 <> x2 and
A4:  F_dp1(t,a).x1 = F_dp1(t,a).x2 by Th27;
A5:  dom (F_dp1(t,a)) = Seg len F_dp1(t,a) by FINSEQ_1:def 3
     .= Seg (t+1) by Def4;
     consider n1 be Nat such that
A6:  x1 = n1 and
A7:  1<= n1 and
A8:  n1 <= t+1 by A1, A5;
     reconsider x1 as Nat by A6;
     consider n2 be Nat such that
A9:  x2 = n2 and
A10: 1 <= n2 and
A11: n2 <= t+1 by A2,A5;
     reconsider x2 as Nat by A9;
     F_dp1(t,a).x1 in rng F_dp1(t,a) by A1,FUNCT_1:3;then
     F_dp1(t,a).x1 in Segm t; then
     F_dp1(t,a).x1 in { i where i is Nat : i < t } by AXIOMS:4; then
     consider i be Nat such that
A13: F_dp1(t,a).x1 = i and i < t;
A14: [\ (frac ((x1 -1)*a))*t /] = i by Def4,A1,A13;
     reconsider r1 = frac ((x1 -1)*a) as Real;
     F_dp1(t,a).x2 in rng F_dp1(t,a) by A2,FUNCT_1:3;then
     F_dp1(t,a).x2 in Segm t; then
     F_dp1(t,a).x2 in { i where i is Nat : i < t } by AXIOMS:4; then
     consider i2 be Nat such that
A16: F_dp1(t,a).x2 = i2 and i2 < t;
A17: [\ (frac ((x2 -1)*a))*t /] = i2 by Def4,A2,A16;
     reconsider r2 = frac ((x2 -1)*a) as Real;
     i = F_dp1(t,a).x1 by A13 .= F_dp1(t,a).x2 by A4 .= i2 by A16; then
A18: r1 in (Equal_Div_interval(t)).i & r2 in (Equal_Div_interval(t)).i
       by A14,Lm4,A17;
A19: |. r1 - r2 .| < t" by Lm5,A18;
     set m1 = x1 - 1;
     set m2 = x2 - 1;
A20: r1 - r2 = m1*a - [\ m1*a /] - frac (m2*a) by INT_1:def 8
     .= m1*a - [\ m1*a /] - (m2*a - [\ m2*a /]) by INT_1:def 8
     .= (m1-m2)*a -( [\ m1*a /] - [\ m2*a /]);
     per cases by A3,XXREAL_0:1;
       suppose
A21:     x1 > x2;
         set x = m1 - m2;
         set y = [\ m1*a /] - [\ m2*a /];
A24:     x = x1 - x2;
A25:     t+1 - x2 >= x1 - x2 by A6,A8,XREAL_1:13;
         t+1 - 1 >= t + 1 - x2 by A9,A10,XREAL_1:10; then
A27:     x <= t by A25,XXREAL_0:2;
A28:     0 < x by A21,XREAL_1:50,A24;
         |. x*a -y .| < t" by Lm5,A18,A20;
         hence thesis by A27,A28;
       end;
       suppose
A29:     x1 < x2;
         set x = m2 - m1;
         set y = [\ m2*a /] - [\ m1*a /];
A32:     x = x2 - x1;
A33:     t+1 - x1 >= x2 - x1 by A9,A11,XREAL_1:13;
         t+1 - 1 >= t + 1 - x1 by A6,A7,XREAL_1:10; then
A35:     x <= t by A33,XXREAL_0:2;
A36:     0 < x by A29,XREAL_1:50,A32;
         -(r1-r2) = x*a - y by A20; then
         |. x*a -y .| < t" by A19,COMPLEX1:52;
         hence thesis by A35,A36;
       end;
     end;
