reserve p, q for Point of TOP-REAL 2,
  r for Real,
  h for non constant standard special_circular_sequence,
  g for FinSequence of TOP-REAL 2,
  f for non empty FinSequence of TOP-REAL 2,
  I, i1, i, j, k for Nat;

theorem Th41:
  for Y being non empty finite Subset of NAT st 1 <= i & i <= len
  f & 1 <= I & I <= len GoB f &
  Y = { j where j is Element of NAT: [I,j] in Indices GoB f & ex k st k in
  dom f & f/.k = (GoB f)*(I,j) } & (f/.i)`1 = ((GoB f)*(I,1))`1 & i1 = min Y
  holds (GoB f)*(I,i1)`2 <= (f/.i)`2
proof
  let Y be non empty finite Subset of NAT;
A1: f/.i=|[(f/.i)`1,(f/.i)`2]| by EUCLID:53;
  assume
A2: 1<=i & i<=len f & 1<=I & I<=len GoB f &
  Y={j where j is Element of NAT:[I,j] in Indices GoB f
  & ex k st k in dom f & f/.k=(GoB f)*(I,j)} & (f/.i)`1=((GoB f)*(I,1))`1 & i1=
  min Y;
  then
A3: i in dom f by FINSEQ_3:25;
  then consider i2,j2 be Nat such that
A4: [i2,j2] in Indices GoB f and
A5: f/.i=(GoB f)*(i2,j2) by GOBOARD5:11;
A6: j2<=width GoB f by A4,MATRIX_0:32;
A7: 1<=j2 by A4,MATRIX_0:32;
  then
A8: [I,j2] in Indices GoB f by A2,A6,MATRIX_0:30;
A9: i2<=len GoB f by A4,MATRIX_0:32;
  1<=i2 by A4,MATRIX_0:32;
  then
A10: (f/.i)`2=((GoB f)*(1,j2))`2 by A5,A9,A7,A6,GOBOARD5:1
    .=((GoB f)*(I,j2))`2 by A2,A7,A6,GOBOARD5:1;
  i1 in Y by A2,XXREAL_2:def 7;
  then ex j being Element of NAT
     st i1=j & [I,j] in Indices GoB f & ex k st k in dom f & f/.k=(
  GoB f)*(I,j) by A2;
  then
A11: 1<=i1 by MATRIX_0:32;
A12: j2 in NAT by ORDINAL1:def 12;
  (f/.i)`1=((GoB f)*(I,j2))`1 by A2,A7,A6,GOBOARD5:2;
  then f/.i=(GoB f)*(I,j2) by A10,A1,EUCLID:53;
  then j2 in Y by A2,A3,A8,A12;
  then
A13: i1<=j2 by A2,XXREAL_2:def 7;
A14: j2<=width GoB f by A4,MATRIX_0:32;
  now
    per cases;
    case
      i1<j2;
      hence (GoB f)*(I,i1)`2<=(GoB f)*(I,j2)`2 by A2,A11,A14,GOBOARD5:4;
    end;
    case
      i1>=j2;
      hence (GoB f)*(I,i1)`2<=(GoB f)*(I,j2)`2 by A13,XXREAL_0:1;
    end;
  end;
  hence thesis by A10;
end;
