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 Th42:
  for Y being non empty finite Subset of NAT st 1 <= i & i <= len
h & 1 <= I & I <= width GoB h &
 Y = { j where j is Element of NAT: [j,I] in Indices GoB h & ex k st k in
  dom h & h/.k = (GoB h)*(j,I) } & (h/.i)`2 = ((GoB h)* (1,I))`2 & i1 = min Y
  holds (GoB h)*(i1,I)`1 <= (h/.i)`1
proof
  let Y be non empty finite Subset of NAT;
A1: h/.i=|[(h/.i)`1,(h/.i)`2]| by EUCLID:53;
  assume
A2: 1<=i & i<=len h & 1 <= I & I <= width GoB h &
Y={j where j is Element of NAT:[j,I] in Indices
GoB h & ex k st k in dom h & h/.k=(GoB h)*(j,I)} & (h/.i)`2=((GoB h)*(1,I))`2 &
  i1=min Y;
  then
A3: i in dom h by FINSEQ_3:25;
  then consider i2,j2 be Nat such that
A4: [i2,j2] in Indices GoB h and
A5: h/.i=(GoB h)*(i2,j2) by GOBOARD5:11;
A6: i2<=len GoB h by A4,MATRIX_0:32;
A7: 1<=i2 by A4,MATRIX_0:32;
  then
A8: [i2,I] in Indices GoB h by A2,A6,MATRIX_0:30;
A9: j2<=width GoB h by A4,MATRIX_0:32;
  1<=j2 by A4,MATRIX_0:32;
  then
A10: (h/.i)`1=((GoB h)*(i2,1))`1 by A5,A7,A6,A9,GOBOARD5:2
    .=((GoB h)*(i2,I))`1 by A2,A7,A6,GOBOARD5:2;
  i1 in Y by A2,XXREAL_2:def 7;
  then ex j being Element of NAT
   st i1=j & [j,I] in Indices GoB h & ex k st k in dom h & h/.k=(
  GoB h)*(j,I) by A2;
  then
A11: 1<=i1 by MATRIX_0:32;
A12: i2 in NAT by ORDINAL1:def 12;
  (h/.i)`2=((GoB h)*(i2,I))`2 by A2,A7,A6,GOBOARD5:1;
  then h/.i=(GoB h)*(i2,I) by A10,A1,EUCLID:53;
  then i2 in Y by A2,A3,A8,A12;
  then
A13: i1<=i2 by A2,XXREAL_2:def 7;
A14: i2<=len GoB h by A4,MATRIX_0:32;
  now
    per cases;
    case
      i1<i2;
      hence (GoB h)*(i1,I)`1<=(GoB h)*(i2,I)`1 by A2,A11,A14,GOBOARD5:3;
    end;
    case
      i1>=i2;
      hence (GoB h)*(i1,I)`1<=(GoB h)*(i2,I)`1 by A13,XXREAL_0:1;
    end;
  end;
  hence thesis by A10;
end;
