
theorem ZMATRLIN43:
  for i being Nat, j being Element of F_Real, k being Element of F_Rat
  st j = k
  holds power(F_Real).(-1_F_Real,i)*j = power(F_Rat).(-1_F_Rat,i)*k
  proof
    let i be Nat,j be Element of F_Real,k be Element of F_Rat;
    assume AS: j = k;
    defpred P[Nat] means
    power(F_Real).(-1_F_Real,$1)*j = power(F_Rat).(-1_F_Rat,$1)*k;
    P1: P[0]
    proof
      power(F_Real).(-1_F_Real,0)*j = 1_(F_Real) *j by GROUP_1:def 7
      .= power(F_Rat).(-1_F_Rat,0)*k by AS,GROUP_1:def 7;
      hence thesis;
    end;
    P2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume AS1: P[n];
      P3: power(F_Real).(-1_F_Real,n+1)*j
      = ((power(F_Real).(-1_F_Real,n)) * (-1_F_Real)) *j by GROUP_1:def 7
      .= (-1_F_Real) * (( power(F_Real).(-1_F_Real,n)) * j);
      power(F_Rat).(-1_F_Rat,n+1)*k
      = ((power(F_Rat).(-1_F_Rat,n)) * (-1_F_Rat)) *k by GROUP_1:def 7
      .= (-1_F_Rat) * (( power(F_Rat).(-1_F_Rat,n)) * k);
      hence power(F_Real).(-1_F_Real,n+1)*j
      = power(F_Rat).(-1_F_Rat,n+1)*k by AS1,P3;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(P1,P2);
    hence thesis;
  end;
