
theorem :: Problem 71 a, for 100
  not ex f being Arithmetic_Progression st
    difference f = 100 &
      ex p1,p2,p3 being Prime, i being Nat st
      p1,p2,p3 are_mutually_distinct &
        p1 = f.i & p2 = f.(i+1) & p3 = f.(i+2)
  proof
    given f being Arithmetic_Progression such that
A1: difference f = 100 & ex p1,p2,p3 being Prime, i be Nat st
      p1,p2,p3 are_mutually_distinct &
        p1 = f.i & p2 = f.(i+1) & p3 = f.(i+2);
    reconsider f as integer-valued Arithmetic_Progression by A1,LemmaIntProg;
    consider p1,p2,p3 being Prime, i be Nat such that
A2: p1,p2,p3 are_mutually_distinct &
        p1 = f.i & p2 = f.(i+1) & p3 = f.(i+2) by A1;
b1: p2 - p1 = difference f by A2,LemmaDiffConst;
    f.(i+1+1) = p3 by A2; then
b2: p3 - p2 = difference f by A2,LemmaDiffConst;
K1: 3 divides p1+1 iff 3 divides p1+1+3*33 by LemmaCong;
    3 divides p1+2 iff 3 divides p1+2+3*66 by LemmaCong; then
    per cases by LemmaDivides,b1,b2,A1,K1,ThreeConsecutive;
    suppose
AA:   p1 = 3;
      203 = 7 * 29; then
      7 divides 203; then
      203 is not prime;
      hence thesis by AA,b1,b2,A1;
    end;
    suppose
      p2 = 3; then
      p1 = 3 - 100 by b1,A1;
      hence thesis by INT_2:def 4;
    end;
    suppose
      p3 = 3; then
      p2 = 3 - 100 by b2,A1; then
      p2 < 0;
      hence thesis;
    end;
  end;
