reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

theorem
  (Sum mlt(R1,R2))^2 <= (Sum sqr R1)*(Sum sqr R2)
proof
A0: i is Nat by TARSKI:1;
  defpred P[Nat] means for R1,R2 being Element of $1-tuples_on REAL holds (Sum
  mlt(R1,R2))^2 <= (Sum sqr R1)*(Sum sqr R2);
A1: for c be Nat st P[c] holds P[c+1]
  proof
    let c be Nat such that
A2: for R1,R2 being Element of c-tuples_on REAL holds (Sum mlt(R1,R2))
    ^2 <= (Sum sqr R1)*(Sum sqr R2);
    let R1,R2 be Element of (c+1)-tuples_on REAL;
    consider R19 being (Element of c-tuples_on REAL),
      x1 being Element of REAL such that
A3: R1 = R19^<*x1*> by FINSEQ_2:117;
    consider R29 being (Element of c-tuples_on REAL),
      x2 being Element of REAL such that
A4: R2 = R29^<*x2*> by FINSEQ_2:117;
A5: for R being Element of c-tuples_on REAL holds
    Sum sqr (R^<*r*>) = Sum sqr R + r^2
    proof
      let R be Element of c-tuples_on REAL;
      reconsider s=r as Element of REAL by XREAL_0:def 1;
      sqr(R^<*s*>) = (sqrreal*R)^<*sqrreal.s*> by FINSEQOP:8
        .= (sqr R)^<*r^2*> by Def2;
      hence thesis by Th74;
    end;
    then
A6: Sum sqr R29 + x2^2 = Sum sqr (R29^<*x2*>)
    .= Sum sqr R2 by A4;
    (Sum mlt(R19,R29))^2 + (0 qua Element of NAT) <= (Sum sqr R19)*(Sum
    sqr R29) by A2;
    then
A7: 0 <= (Sum sqr R19)*(Sum sqr R29) - (Sum mlt(R19,R29))^2 by XREAL_1:19;
    mlt(R19^<*x1*>,R29^<*x2*>) = (multreal.:(R19,R29))^<*multreal.(x1,x2)
    *> by FINSEQOP:10
      .= (mlt(R19,R29))^<*x1*x2*> by BINOP_2:def 11;
    then
A8: Sum mlt(R19^<*x1*>,R29^<*x2*>) = Sum mlt(R19,R29) + x1*x2 by Th74;
A9: 2*(x1*x2)*Sum mlt(R19,R29) = 2*((x1*x2)*Sum mlt(R19,R29))
      .= 2*Sum((x1*x2)*mlt(R19,R29)) by Th87
      .= 2*Sum(x1*(x2*mlt(R19,R29))) by RFUNCT_1:17
      .= 2*Sum(x1*mlt(R29,x2*R19)) by RFUNCT_1:12
      .= 2*Sum(mlt(x1*R29,x2*R19)) by FINSEQOP:26;
A10: -(Sum mlt(R19,R29)+x1*x2)^2 = -(x1*x2)^2+-(2*(x1*x2)*Sum mlt(R19,R29)
    +(Sum mlt(R19,R29))^2)
      .= -x1^2*x2^2+(-(Sum mlt(R19,R29))^2+ -2*Sum(mlt(x1*R29,x2*R19))) by A9;
A11: 0 <= Sum sqr (x1*R29-x2*R19) by Th86;
A12: (Sum sqr R19 + x1^2)*(Sum sqr R29 + x2^2) = (Sum sqr R19)*(Sum sqr R29
    )+(x1^2*(Sum sqr R29)+(Sum sqr R19)*x2^2)+x1^2*x2^2
      .= (Sum sqr R19)*(Sum sqr R29)+(Sum(x1^2*sqr R29)+(Sum sqr R19)*x2^2)+
    x1^2*x2^2 by Th87
      .= (Sum sqr R19)*(Sum sqr R29)+(Sum sqr(x1*R29)+x2^2*(Sum sqr R19))+x1
    ^2*x2^2 by Th58
      .= (Sum sqr R19)*(Sum sqr R29)+(Sum sqr(x1*R29)+Sum (x2^2*sqr R19))+x1
    ^2*x2^2 by Th87
      .= (Sum sqr R19)*(Sum sqr R29)+(Sum sqr(x1*R29)+Sum sqr(x2*R19))+x1^2*
    x2^2 by Th58;
A13: Sum sqr(x1*R29)+Sum sqr(x2*R19) + -2*Sum(mlt(x1*R29,x2*R19)) = Sum
    sqr(x1*R29)-2*Sum(mlt(x1*R29,x2*R19))+Sum sqr(x2*R19)
      .= Sum sqr(x1*R29)-Sum(2*mlt(x1*R29,x2*R19))+Sum sqr(x2*R19) by Th87
      .= Sum(sqr(x1*R29)-2*mlt(x1*R29,x2*R19))+Sum sqr(x2*R19) by Th90
      .= Sum(sqr (x1*R29)-2*mlt(x1*R29,x2*R19)+sqr(x2*R19)) by Th89;
    (Sum sqr R19 + x1^2) = Sum sqr R1 by A3,A5;
    then
    (Sum sqr R1)*(Sum sqr R2) - (Sum mlt(R1,R2))^2 = (Sum sqr R19 + x1^2)
    *(Sum sqr R29 + x2^2) + -(Sum mlt(R19,R29)+x1*x2)^2 by A3,A4,A8,A6
      .= (Sum sqr R19)*(Sum sqr R29)+-(Sum mlt(R19,R29))^2 + ((Sum sqr(x1*
    R29)+Sum sqr(x2*R19)) +- 2*Sum(mlt(x1*R29,x2*R19))) by A12,A10
      .= (Sum sqr R19)*(Sum sqr R29)-(Sum mlt(R19,R29))^2 + Sum sqr (x1*R29-
    x2*R19) by A13,Th69;
    then (Sum mlt(R1,R2))^2 + (0 qua Element of NAT) <= (Sum sqr R1)*(Sum sqr
    R2) by A7,A11,XREAL_1:19;
    hence thesis;
  end;
A14: P[0];
  for i be Nat holds P[i] from NAT_1:sch 2(A14,A1);
  hence thesis by A0;
end;
